47 operations of 2nd degree expressions and formulas

Special Binomial Operations

A binomial is a two-term polynomial. Special Binomial Operations

A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.

Special Binomial Operations

A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.A trinomial is a three term polynomial.

Special Binomial Operations

A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c.

Special Binomial Operations

A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c.The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + #

Special Binomial Operations

A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c.The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + #

Special Binomial Operations

F: To get the x2-term, multiply the two Front x-terms of the binomials.

A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c.The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + #

Special Binomial Operations

F: To get the x2-term, multiply the two Front x-terms of the binomials.OI: To get the x-term, multiply the Outer and Inner pairs and combine the results.

A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c.The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + #

Special Binomial Operations

F: To get the x2-term, multiply the two Front x-terms of the binomials.OI: To get the x-term, multiply the Outer and Inner pairs and combine the results.L: To get the constant term, multiply the two Last constant terms.

A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c.The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + #

Special Binomial Operations

F: To get the x2-term, multiply the two Front x-terms of the binomials.OI: To get the x-term, multiply the Outer and Inner pairs and combine the results.L: To get the constant term, multiply the two Last constant terms. This is called the FOIL method.

A binomial is a two-term polynomial. Usually we use the term for expressions of the form ax + b.A trinomial is a three term polynomial. Usually we use the term for expressions of the form ax2 + bx + c.The product of two binomials is a trinomial. (#x + #)(#x + #) = #x2 + #x + #

Special Binomial Operations

F: To get the x2-term, multiply the two Front x-terms of the binomials.OI: To get the x-term, multiply the Outer and Inner pairs and combine the results.L: To get the constant term, multiply the two Last constant terms. This is called the FOIL method.The FOIL method speeds up the multiplication of above binomial products and this will come in handy later.

Example A. Multiply using FOIL method.a. (x + 3)(x – 4)

Special Binomial Operations

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2

Special Binomial Operations

The front terms: x2-term

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2

Special Binomial Operations

Outer pair: –4x

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2

Special Binomial Operations

Inner pair: –4x + 3x

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x

Special Binomial Operations

Outer Inner pairs: –4x + 3x = –x

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

Special Binomial Operations

The last terms: –12

Special Binomial Operations

b. (3x + 4)(–2x + 5)

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: –12

Special Binomial Operations

b. (3x + 4)(–2x + 5) = –6x2

The front terms: –6x2

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: –12

Special Binomial Operations

b. (3x + 4)(–2x + 5) = –6x2

Outer pair: 15x

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: –12

Special Binomial Operations

b. (3x + 4)(–2x + 5) = –6x2

Inner pair: 15x – 8x

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: –12

Special Binomial Operations

b. (3x + 4)(–2x + 5) = –6x2 + 7x

Outer and Inner pair: 15x – 8x = 7x

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: –12

Special Binomial Operations

b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: 20

The last terms: –12

Special Binomial Operations

b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: 20

The last terms: –12

Expanding the negative of the binomial product requires extra care.

Special Binomial Operations

b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: 20

The last terms: –12

Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.

Special Binomial Operations

b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: 20

The last terms: –12

Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.Example B. Expand.a. – (3x – 4)(x + 5)

Special Binomial Operations

b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: 20

The last terms: –12

Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.Example B. Expand.a. – [(3x – 4)(x + 5)] Insert [ ]

Special Binomial Operations

b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: 20

The last terms: –12

Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.Example B. Expand.a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20]

Insert [ ]Expand

Special Binomial Operations

b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: 20

The last terms: –12

Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.Example B. Expand.a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20] = – [ 3x2 + 11x – 20]

Insert [ ]Expand

Special Binomial Operations

b. (3x + 4)(–2x + 5) = –6x2 + 7x + 20

Example A. Multiply using FOIL method.a. (x + 3)(x – 4) = x2 – x – 12

The last terms: 20

The last terms: –12

Expanding the negative of the binomial product requires extra care. One way to do this is to insert a set of “[ ]” around the product.Example B. Expand.a. – [(3x – 4)(x + 5)] = – [ 3x2 + 15x – 4x – 20] = – [ 3x2 + 11x – 20] = – 3x2 – 11x + 20

Insert [ ]Expand

Remove [ ] andchange signs.

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5)

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) Distribute the sign.

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20

Distribute the sign.Expand

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20

Distribute the sign.Expand

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20

Distribute the sign.Expand

b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20

Distribute the sign.Expand

b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)= (2x – 5)(x +3) + (–3x + 4)(x + 5)

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20

Distribute the sign.Expand

b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)= (2x – 5)(x +3) + (–3x + 4)(x + 5)= 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20)

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20

Distribute the sign.Expand

b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)= (2x – 5)(x +3) + (–3x + 4)(x + 5)= 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20)= 2x2 + x – 15 – 3x2 – 11x + 20

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20

Distribute the sign.Expand

b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)= (2x – 5)(x +3) + (–3x + 4)(x + 5)= 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20)= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20

Distribute the sign.Expand

b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)= (2x – 5)(x +3) + (–3x + 4)(x + 5)= 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20)= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5 (2x – 5)(x +3) – (3x – 4)(x + 5)

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20

Distribute the sign.Expand

b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)= (2x – 5)(x +3) + (–3x + 4)(x + 5)= 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20)= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20

Distribute the sign.Expand

b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)= (2x – 5)(x +3) + (–3x + 4)(x + 5)= 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20)= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)]= 2x2 + x – 15 – [3x2 +11x – 20]

Insert bracketsExpand

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20

Distribute the sign.Expand

b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)= (2x – 5)(x +3) + (–3x + 4)(x + 5)= 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20)= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets= 2x2 + x – 15 – [3x2 +11x – 20]= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5

ExpandRemove brackets and combine

Special Binomial OperationsAnother way to do this is to distribute the negative sign into the first binomial then FOIL.Example C. Expand.a. – (3x – 4)(x + 5) = (–3x + 4)(x + 5) = – 3x2 – 15x + 4x + 20 = – 3x2 – 11x + 20

Distribute the sign.Expand

b. Expand and simplify. (Two versions) (2x – 5)(x +3) – (3x – 4)(x + 5)= (2x – 5)(x +3) + (–3x + 4)(x + 5)= 2x2 + 6x – 5x – 15 + (–3x2 –15x + 4x + 20)= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5 (2x – 5)(x +3) – [(3x – 4)(x + 5)] Insert brackets= 2x2 + x – 15 – [3x2 +11x – 20]= 2x2 + x – 15 – 3x2 – 11x + 20= –x2 – 10x + 5

ExpandRemove brackets and combine

Special Binomial OperationsIf the binomials are in x and y, then the products consist of the x2, xy and y2 terms.

Special Binomial OperationsIf the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is,

(#x + #y)(#x + #y) = #x2 + #xy + #y2

Special Binomial OperationsIf the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is,

(#x + #y)(#x + #y) = #x2 + #xy + #y2

The FOIL method is still applicable in this case.

Special Binomial OperationsIf the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is,

Example D. Expand.(3x – 4y)(x + 5y)

(#x + #y)(#x + #y) = #x2 + #xy + #y2

The FOIL method is still applicable in this case.

Special Binomial OperationsIf the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is,

Example D. Expand.(3x – 4y)(x + 5y)= 3x2

(#x + #y)(#x + #y) = #x2 + #xy + #y2

The FOIL method is still applicable in this case.

F

Special Binomial OperationsIf the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is,

Example D. Expand.(3x – 4y)(x + 5y)= 3x2 + 15xy – 4yx

(#x + #y)(#x + #y) = #x2 + #xy + #y2

The FOIL method is still applicable in this case.

F OI

Special Binomial OperationsIf the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is,

Example D. Expand.(3x – 4y)(x + 5y)= 3x2 + 15xy – 4yx – 20y2

(#x + #y)(#x + #y) = #x2 + #xy + #y2

The FOIL method is still applicable in this case.

F OI L

Special Binomial OperationsIf the binomials are in x and y, then the products consist of the x2, xy and y2 terms. That is,

Example D. Expand.(3x – 4y)(x + 5y)= 3x2 + 15xy – 4yx – 20y2

= 3x2 + 11xy – 20y2

(#x + #y)(#x + #y) = #x2 + #xy + #y2

The FOIL method is still applicable in this case.

Multiplication Formulas

There are some important patterns in multiplying expressions that it is worthwhile to memorize.

Multiplication Formulas

The two binomials (A + B) and (A – B) are said to be the conjugate of each other.

There are some important patterns in multiplying expressions that it is worthwhile to memorize.

Multiplication Formulas

The two binomials (A + B) and (A – B) are said to be the conjugate of each other. For example, the conjugate of (3x + 2) is (3x – 2),

There are some important patterns in multiplying expressions that it is worthwhile to memorize.

Multiplication Formulas

The two binomials (A + B) and (A – B) are said to be the conjugate of each other. For example, the conjugate of (3x + 2) is (3x – 2), andthe conjugate of (2ab – c) is (2ab + c).

There are some important patterns in multiplying expressions that it is worthwhile to memorize.

Multiplication Formulas

The two binomials (A + B) and (A – B) are said to be the conjugate of each other. For example, the conjugate of (3x + 2) is (3x – 2), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).

There are some important patterns in multiplying expressions that it is worthwhile to memorize.

Multiplication Formulas

The two binomials (A + B) and (A – B) are said to be the conjugate of each other.

There are some important patterns in multiplying expressions that it is worthwhile to memorize.

I. Difference of Squares Formula

Multiplication Formulas

For example, the conjugate of (3x + 2) is (3x – 2), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).

The two binomials (A + B) and (A – B) are said to be the conjugate of each other.

There are some important patterns in multiplying expressions that it is worthwhile to memorize.

I. Difference of Squares Formula (A + B)(A – B)

Conjugate Product

Multiplication Formulas

For example, the conjugate of (3x + 2) is (3x – 2), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).

The two binomials (A + B) and (A – B) are said to be the conjugate of each other.

There are some important patterns in multiplying expressions that it is worthwhile to memorize.

I. Difference of Squares Formula (A + B)(A – B) = A2 – B2

Conjugate Product Difference of Squares

Multiplication Formulas

For example, the conjugate of (3x + 2) is (3x – 2), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).

The two binomials (A + B) and (A – B) are said to be the conjugate of each other.

There are some important patterns in multiplying expressions that it is worthwhile to memorize.

I. Difference of Squares Formula (A + B)(A – B) = A2 – B2

To verify this :(A + B)(A – B)

Conjugate Product Difference of Squares

Multiplication Formulas

For example, the conjugate of (3x + 2) is (3x – 2), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).

The two binomials (A + B) and (A – B) are said to be the conjugate of each other.

There are some important patterns in multiplying expressions that it is worthwhile to memorize.

I. Difference of Squares Formula (A + B)(A – B) = A2 – B2

To verify this :(A + B)(A – B) = A2 – AB + AB – B2

Conjugate Product Difference of Squares

Multiplication Formulas

For example, the conjugate of (3x + 2) is (3x – 2), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).

The two binomials (A + B) and (A – B) are said to be the conjugate of each other.

There are some important patterns in multiplying expressions that it is worthwhile to memorize.

I. Difference of Squares Formula (A + B)(A – B) = A2 – B2

To verify this :(A + B)(A – B) = A2 – AB + AB – B2

= A2 – B2

Conjugate Product Difference of Squares

Multiplication Formulas

For example, the conjugate of (3x + 2) is (3x – 2), andthe conjugate of (2ab – c) is (2ab + c).Note: The conjugate is different from the opposite. The opposite of (3x + 2) is (–3x – 2).

Multiplication FormulasHere are some examples of squaring:

Multiplication FormulasHere are some examples of squaring: (3x)2 =

Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2,

Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2, (2xy)2 =

Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2,

Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2

Multiplication FormulasHere are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2)

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2)

(A + B)(A – B)

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2

(A + B)(A – B) = A2 – B2

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2)

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2

= 4x2y2 – 25z4

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2

= 4x2y2 – 25z4

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

II. Square Formulas

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2

= 4x2y2 – 25z4

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

II. Square Formulas (A + B)2 = A2 + 2AB + B2

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2

= 4x2y2 – 25z4

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

II. Square Formulas (A + B)2 = A2 + 2AB + B2

(A – B)2 = A2 – 2AB + B2

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2

= 4x2y2 – 25z4

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

II. Square Formulas (A + B)2 = A2 + 2AB + B2

(A – B)2 = A2 – 2AB + B2

We may check this easily by multiplying,

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2

= 4x2y2 – 25z4

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

II. Square Formulas (A + B)2 = A2 + 2AB + B2

(A – B)2 = A2 – 2AB + B2

We may check this easily by multiplying,(A + B)2 = (A + B)(A + B)

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2

= 4x2y2 – 25z4

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

II. Square Formulas (A + B)2 = A2 + 2AB + B2

(A – B)2 = A2 – 2AB + B2

We may check this easily by multiplying,(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2

= 4x2y2 – 25z4

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

II. Square Formulas (A + B)2 = A2 + 2AB + B2

(A – B)2 = A2 – 2AB + B2

We may check this easily by multiplying,(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2

= 4x2y2 – 25z4

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

II. Square Formulas (A + B)2 = A2 + 2AB + B2

(A – B)2 = A2 – 2AB + B2

We may check this easily by multiplying,(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2

We say that “(A + B)2 is A2, B2, plus twice A*B”,

Multiplication Formulas

Example E. Expand.a. (3x + 2)(3x – 2) = (3x)2 – (2)2 = 9x2 – 4

(A + B)(A – B) = A2 – B2 b. (2xy – 5z2)(2xy + 5z2) = (2xy)2 – (5z2)2

= 4x2y2 – 25z4

Here are some examples of squaring: (3x)2 = 9x2, (2xy)2 = 4x2y2, and (5z2)2 = 25z4.

II. Square Formulas (A + B)2 = A2 + 2AB + B2

(A – B)2 = A2 – 2AB + B2

We may check this easily by multiplying,(A + B)2 = (A + B)(A + B) = A2 + AB + BA + B2 = A2 + 2AB + B2

We say that “(A + B)2 is A2, B2, plus twice A*B”, and “(A – B)2 is A2, B2, minus twice A*B”.

Example F.a. (3x + 4)2

Multiplication Formulas

Example F.a. (3x + 4)2

(A + B)2

Multiplication Formulas

Example F.a. (3x + 4)2

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

Example F.a. (3x + 4)2 = (3x)2

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4)

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the Formulas

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the FormulasWe can use the above formulas to help us multiply.

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the FormulasWe can use the above formulas to help us multiply.Example G. Calculate. Use the conjugate formula.a. 51*49

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the FormulasWe can use the above formulas to help us multiply.Example G. Calculate. Use the conjugate formula.a. 51*49 = (50 + 1)(50 – 1)

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the FormulasWe can use the above formulas to help us multiply.Example G. Calculate. Use the conjugate formula.a. 51*49 = (50 + 1)(50 – 1) = 502 – 12

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the FormulasWe can use the above formulas to help us multiply.Example G. Calculate. Use the conjugate formula.a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the FormulasWe can use the above formulas to help us multiply.Example G. Calculate. Use the conjugate formula.a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499b. 52*48

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the FormulasWe can use the above formulas to help us multiply.Example G. Calculate. Use the conjugate formula.a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499b. 52*48 = (50 + 2)(50 – 2) = 502 – 22

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the FormulasWe can use the above formulas to help us multiply.Example G. Calculate. Use the conjugate formula.a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the FormulasWe can use the above formulas to help us multiply.Example G. Calculate. Use the conjugate formula.a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496c. 63*57 =

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the FormulasWe can use the above formulas to help us multiply.Example G. Calculate. Use the conjugate formula.a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496c. 63*57 = (60 + 3)(60 – 3) = 602 – 32

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the FormulasWe can use the above formulas to help us multiply.Example G. Calculate. Use the conjugate formula.a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591

Example F.a. (3x + 4)2 = (3x)2 + 2(3x)(4) + 42 = 9x2 + 24x + 16

(A + B)2 = A2 + 2AB + B2

Multiplication Formulas

b. (3a – 5b)2 = (3a)2 – 2(3a)(5b) + (5b)2

= 9a2 – 30ab + 25b2

III. Some Applications of the FormulasWe can use the above formulas to help us multiply.

The conjugate formula (A + B)(A – B) = A2 – B2

may be used to multiply two numbers of the forms(A + B) and (A – B) where A2 and B2 can be calculated easily.

Example G. Calculate. Use the conjugate formula.a. 51*49 = (50 + 1)(50 – 1) = 502 – 12 = 2,500 – 1 = 2,499b. 52*48 = (50 + 2)(50 – 2) = 502 – 22 = 2,500 – 4 = 2,496c. 63*57 = (60 + 3)(60 – 3) = 602 – 32 = 3,600 – 9 = 3,591

Multiplication Formulas

We observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2

(1 – r)(1 + r + r2) = 1 – r3

(1 – r)(1 + r + r2 + r3) = 1 – r4

(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5

...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rn

The Telescoping Products

These are telescoping products, the products compress into two terms. In particular, we get the sum–of–powers formula:

(1 – r)(1 + r + r2 … + rn-1) = 1 – rn

1 – r

Exercise. A. Calculate. Use the conjugate formula.Multiplication Formulas

1. 21*19 2. 31*29 3. 41*39 4. 71*69 5. 22*18 6. 32*28 7. 52*48 8. 73*67 B. Calculate. Use the squaring formula.9. 212 10. 312 11. 392 12. 692 13. 982 14. 30½2

15. 100½2 16. 49½2

18. (x + 5)(x – 5) 19. (x – 7)(x + 7)20. (2x + 3)(2x – 3) 21. (3x – 5)(3x + 5)

C. Expand.

22. (7x + 2)(7x – 2) 23. (–7 + 3x )(–7 – 3x)24. (–4x + 3)(–4x – 3) 25. (2x – 3y)(2x + 3y)26. (4x – 5y)(5x + 5y) 27. (1 – 7y)(1 + 7y)28. (5 – 3x)(5 + 3x) 29. (10 + 9x)(10 – 9x)30. (x + 5)2 31. (x – 7)2

32. (2x + 3)2 33. (3x + 5y)2

34. (7x – 2y)2 35. (2x – h)2

B. Expand and simplify.

Special Binomial Operations

1. (x + 5)(x + 7) 2. (x – 5)(x + 7)3. (x + 5)(x – 7) 4. (x – 5)(x – 7)5. (3x – 5)(2x + 4) 6. (–x + 5)(3x + 8)7. (2x – 5)(2x + 5) 8. (3x + 7)(3x – 7)

Exercise. A. Expand by FOIL method first. Then do them by inspection.

9. (–3x + 7)(4x + 3) 10. (–5x + 3)(3x – 4)11. (2x – 5)(2x + 5) 12. (3x + 7)(3x – 7)13. (9x + 4)(5x – 2) 14. (–5x + 3)(–3x + 1)15. (5x – 1)(4x – 3) 16. (6x – 5)(–2x + 7)17. (x + 5y)(x – 7y) 18. (x – 5y)(x – 7y)19. (3x + 7y)(3x – 7y) 20. (–5x + 3y)(–3x + y)

21. –(2x – 5)(x + 3) 22. –(6x – 1)(3x – 4)23. –(8x – 3)(2x + 1) 24. –(3x – 4)(4x – 3)

C. Expand and simplify.25. (3x – 4)(x + 5) + (2x – 5)(x + 3)26. (4x – 1)(2x – 5) + (x + 5)(x + 3)27. (5x – 3)(x + 3) + (x + 5)(2x – 5)

Special Binomial Operations

28. (3x – 4)(x + 5) – (2x – 5)(x + 3)29. (4x – 4)(2x – 5) – (x + 5)(x + 3)30. (5x – 3)(x + 3) – (x + 5)(2x – 5)31. (2x – 7)(2x – 5) – (3x – 1)(2x + 3)32. (3x – 1)(x – 7) – (x – 7)(3x + 1)33. (2x – 3)(4x + 3) – (x + 2)(6x – 5)34. (2x – 5)2 – (3x – 1)2

35. (x – 7)2 – (2x + 3)2

36. (4x + 3)2 – (6x – 5)2

Ex. D. Multiply the following monomials.1. 3x2(–3x2)

11. 4x(3x – 5) – 9(6x – 7)

Polynomial Operations

2. –3x2(8x5) 3. –5x2(–3x3)

4. –12( ) 6–5x3

5. 24( x3) 8–5 6. 6x2( ) 3

2x3

7. –15x4( x5) 5–2

F. Expand and simplify.

E. Fill in the degrees of the products. 8. #x(#x2 + # x + #) = #x? + #x? + #x? 9. #x2(#x4 + # x3 + #x2) = #x? + #x? + #x? 10. #x4(#x3 + # x2 + #x + #) = #x? + #x? + #x? + #x?

12. –x(2x + 7) + 3(4x – 2)13. –3x(3x + 2) – 8x(7x – 5) 14. 5x(–5x + 9) + 6x(6x – 1)15. 2x(–4x + 2) – 3x(2x – 1) – 3(4x – 2)16. –4x(–7x + 9) – 2x(2x – 5) + 9(4x + 2)

18. (x + 5)(x + 7)

Polynomial OperationsG. Expand and simplify. (Use any method.)

19. (x – 5)(x + 7)20. (x + 5)(x – 7) 21. (x – 5)(x – 7)22. (3x – 5)(2x + 4) 23. (–x + 5)(3x + 8)24. (2x – 5)(2x + 5) 25. (3x + 7)(3x – 7)26. (3x2 – 5)(x – 6) 27. (8x – 2)(–4x2 – 7)28. (2x – 7)(x2 – 3x + 9) 29. (5x + 3)(2x2 – x + 5)

38. (x – 1)(x + 1) 39. (x – 1)(x2 + x + 1)40. (x – 1)(x3 + x2 + x + 1)41. (x – 1)(x4 + x3 + x2 + x + 1)42. What do you think the answer is for (x – 1)(x50 + x49 + …+ x2 + x + 1)?

30. (x – 1)(x – 1) 31. (x + 1)2

32. (2x – 3)2 33. (5x + 4)2

34. 2x(2x – 1)(3x + 2) 35. 4x(3x – 2)(2x + 3)36. (x – 5)(2x – 1)(3x + 2) 37. (2x + 1)(3x + 1)(x – 2)