chapter 6 section 6.1-factoring out the greatest common...

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Province – Mathematics Department – Southwest Tennessee Community College

Chapter 6

Section 6.1-Factoring out the Greatest Common Factor (GCF) of a Polynomial Objectives:

1. Find the greatest common factor of a list of numbers. 2. Find the greatest common factor of a list of variable

terms. 3. Factor out the greatest common factor. 4. Factor by grouping.

The greatest common factor (GCF) of a list of integers is the largest common factor of those integers. This means 6 is the greatest common factor of 18 and 24, since it is the largest of their common factors. Note Factors of a number are also divisors of the number. The greatest common factor is the same as the greatest common divisor. Ex. Find the greatest common factor for each list of numbers. 36, 60 18, 90, 126 48, 61, 72

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Finding the Greatest Common Factor for Variable Terms

Note: The exponent on a variable in the GCF is the least exponent that appears on that variable in all the terms. Ex. Find the greatest common factor for each list of terms. 12x

2, –30x

5

–x

5y

2, –x

4y

3, –x

8y

6, –x

7

Note In a list of negative terms, sometimes a negative common factor is preferable (even though it is not the greatest common factor). In (b) above, we might prefer –x

4

as the common factor. Finding the Greatest Common Factor (GCF) Step 1: Factor. Write each number in prime factored form. Step 2 List common factors. List each prime number or each variable that is a factor of every term in the list. (If a prime does not appear in one of the prime factored forms, it cannot appear in the greatest common factor.) Step 3 Choose least exponents. Use as exponents on the common prime factors the least exponents from the prime factored forms. Step 4 Multiply. Multiply the primes from Step 3. If there are no primes left after Step 3, the greatest common factor is 1.

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CAUTION The polynomial 3m + 12 is not in factored form when written as the sum 3 · m + 3 · 4. Not in factored form The terms are factored, but the polynomial is not. The factored form of 3m + 12 is the product 3(m + 4). In factored form Ex. Find the Greatest Common Factor (GCF)

xyy 42

2322 14357 xxyx

210525 xx

xyyxxy 26213 22

32 3621 xyxx

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Note: Whenever we factor a polynomial in which the coefficient of the first term is negative, we will factor out the negative common factor, even if it is just – 1. CAUTION: Be sure to include the 1 in a problem like Example 3(b). Check that the factored form can be multiplied out to give the original polynomial.

Factoring cbxax 2 (AC METHOD / GROUPING)

Factor out the GCF.

Multiply the coefficients of the 1st and last terms of

the trinomial.

Find factors of the product from step 2 whose sum is equal to the coefficient of the middle term of the trinomial.

If the product from step 2 is positive, both factors will have the same sign as the middle term.

If the product from step 2 is negative, the larger factor will have the same sign as the middle term.

Rewrite the middle term using the two factors found in step 3.

Factor by grouping. . . 1. Insert a BIG + to separate the first 2 terms

from the last 2 terms. 2. Factor out the GCF in the 1

st 2 terms.

3. Factor out the GCF in the 2nd

2 terms. 4. Factor out the resulting common factor if

possible. (Remember that you may have to factor out a negative.)

CHECK (especially your signs) using FOIL!!!

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Ex. First let’s look at factor by grouping

1234 yxxy

10235 2 xx

45 2 xx

8557 2 xx

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Section 6.2 – Factoring Trinomials Objectives:

1. Factors trinomials with a coefficient of 1 for the squared term.

2. Factor trinomials after factoring out the greatest common factor.

Factoring cbxax 2

(REVERSE FOIL METHOD / TRIAL-AND-ERROR)

Arrange in decreasing power of x .

Factor out a -1 if the first term is negative.

Factor by trial-and-error (REVERSE FOIL) as the product of two binomials if possible.

NOTE: If c is POSITIVE, the signs of the last terms of each binomial will be the SAME. (Look for a SUM.) If c is NEGATIVE, the signs of the last terms of the binomials will be DIFFERENT. (Look for a DIFFERENCE.)

CHECK (especially your signs) using FOIL!!!

Factors of ‘c’ that add to give you ‘b’. Same sign both positive.

Factors of ‘c’ that subtract to give you ‘b’. Larger value gets the sign of ‘b’ (plus sign).

Factors of ‘c’ that subtract to give you ‘b’. Larger value gets the sign of ‘b’ (minus sign).

Factors of ‘c’ that add to give you ‘b’. Same sign both negative.

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Ex. Factor the trinomials

16102 xx

28112 xx

27122 xx

1522 xx

1452 yy

772 pp

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Section 6.3 - Factoring Trinomials by Grouping Objectives:

1. Factor trinomials by grouping when the coefficient of the squared term is not 1.

To Factor when a≠1 Step 1: Multiply a and c. Step 2: Find factors of ac that add/subtract to give you b. Step 3: Rewrite the middle term (bx) with your two factors. (Remember to put the variable on the coefficients) Now you have four terms. Step 4: Group the first two terms together and the second two terms together. Step 5: Factor out the GCF of the first two terms and the last two terms. Step 6: Make sure you binomials match. If not something went wrong. Step 7: Write down the binomial that matches once then group the items on the outside together.

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Factors of ‘a*c’ that add to give you ‘b’.

Factors of ‘a*c’ that subtract to give you ‘b’. Larger value gets the sign of ‘b’ (plus sign).

Factors of ‘a*c’ that subtract to give you ‘b’. Larger value gets the sign of ‘b’ (minus sign).

Factors of ‘a*c’ that add to give you ‘b’.

With leading coefficient 1

11232 2 xx

3633 2 yy

9465 2 rr

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4277 2 pp

143112 2 xx

2068 2 xx

ppp 8557 23

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Section 6.4 - Factoring Trinomials Using FOIL Objectives: 1. Factor trinomials using FOIL. Note:If the original polynomial has no common factor, then none of its binomial factors will either.

So the solution is

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So the solution is

So the solution is

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Section 6.5 - Special Factoring Techniques Objectives: 1. Factor a difference of squares. 2. Factor a perfect square trinomial. 3. Factor a difference of cubes. 4. Factor a sum of cubes.

Perfect Square Trinomials

222

222

2(

2

AAXXAX

AAXXAX

2

4x

2

82x

2323 xx

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THE FOIL METHOD (A SHORTCUT FOR MULTIPLYING TWO BINOMIALS) Multiply the “First terms”, “Outside terms”, “Inside terms”, and the “Last terms” and then “combine like terms”.

1253 xx

1258 xx

282 xx

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Two Special Factorizations – Difference of Squares (Binomials) and Perfect Squares (Trinomials) These are two special factorization: Difference of Squares

))((22 bababa

Perfect Square Trinomials

222 )())((2 baorbabababa

222 )())((2 baorbabababa

169 2 x

149 2 y

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21227 x

254 2 x

22 168 yxyx

49168144 2 xx

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2164025 yy

42625 2 yy

Ex. Factor the difference of cubes.

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Ex. Factor the sum of cubes.

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Section 6.6 - A General Approach to Factoring Objectives:

1. Factor out any common factor. 2. Factor binomials. 3. Factor trinomials. 4. Factor polynomials with more than three terms.

Factoring a Polynomial A polynomial is completely factored when

1) it is written as a product of prime polynomials with integer coefficients, and

2) none of the of polynomial factors can be factored further.

Factoring Out a Common Factor This step is always the same, regardless of the number of terms in the polynomial. Factor each polynomial.

(a) (b)

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Factoring Binomials

Use one of the rules to factor each binomial if possible.

(a) (b)

(c) (d)

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Factoring Trinomials

Factor each trinomial

(a) (b)

(c) (d)

(e)

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Factor each polynomial. Consider factoring by grouping

(a)

(b)

(c)

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Section 6.7 - Solving Quadratic Equations by Factoring Objectives:

1. Solve quadratic equations by factoring. 2. Solve other equations by factoring.

Quadratic Equations A quadratic equation is an equation that can be written in the form ax

2 + bx + c = 0, where a, b, and c, are real

numbers, with a ≠ 0. The given form is called standard form. Zero-Factor Property If a and b are real numbers and ab = 0, then a = 0 or b = 0. In words, if the product of two numbers is 0, then at least one of the numbers must be 0. One number must be 0, but both may be 0. Example: Solve the equation (2y – 3)(y + 1) = 0 Example: Solve the equation x

2 – 3x = 4

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Solving a Quadratic Equation by Factoring Step 1 Write the equation in standard form, that is, with all terms on one side of the equals sign in descending powers of the variable and 0 on the other side. Step 2 Factor completely. Step 3 Use the zero-factor property to set each factor with a variable equal to 0. Step 4 Solve the resulting equations. Step 5 Check each solution in the original equation. Example: Solve the equation 2x

2 + 30 = –16x

y(2y + 5) = 42

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x2 = 4x

x

2 + 64 = –16x

3a

3 – 48a = 0

Note Not all quadratic equations can be solved by factoring

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Section 6.8 - Applications of Quadratic Equations Objectives:

1. Solve problems about geometric figures. 2. Solve problems about consecutive integers. 3. Solve problems using the Pythagorean formula. 4. Solve problems using given quadratic models.

Solving an Applied Problem Step 1 Read the problem, several times if necessary, until you understand what is given and what is to be found. Step 2 Assign a variable to represent the unknown value, using diagrams or tables as needed. Write a statement that tells what the variable represents. Express any other unknown values in terms of the variable. Step 3 Write an equation using the variable expression(s). Step 4 Solve the equation. Step 5 State the answer. Does it seem reasonable? Step 6 Check the answer in the words of the original problem.

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Example: Amy designs silver jewelry. Currently she is working on a triangular pendant. She prefers that the height be twice the base. She has enough silver to make the area of the triangle 4 in

2. What will the dimensions of her pendant

be?

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Problem-Solving Hint In consecutive integer problems, if x represents the first integer, then for two consecutive integers, use x, x+1; three consecutive integers, use x, x+1, x+2; two consecutive even or odd integers, use x, x+2; three consecutive even or odd integers, use x, x+2, x+4. Example: John opened his phonebook to view the city map, which was printed across two adjacent pages. John noticed that the product of these two page numbers was 812. On what two pages was the city map printed?

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Pythagorean Formula If a right triangle (a triangle with a 90° angle) has longest side of length c and two other sides of length a and b, then a

2 + b

2 = c

2.

The longest side, the hypotenuse, is opposite the right angle. The two shorter sides are the legs of the triangle. Ex: Carrie and Diego left Oahu at the same time. Diego sailed directly north, and Carrie sailed directly west. At the exact same time, they both dropped anchor to fish. At that moment, Carrie was 20 miles further from Hawaii than Diego was, and the distance between them was 100 miles. How far from Oahu were Carrie and Diego?

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Ex: A pebble falls off the roof of a 560 ft building according to the quadratic equation h = –16t

2 + 560, where h = the height

of the Pebble and t = the time in seconds. After how many seconds will the pebble be 160 ft above the ground?

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