unit essential questions are two algebraic expressions
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Unit Essential Questions
Are two algebraic expressions that appear to be different actually equivalent?
What is the relationship between properties of real numbers and
properties of polynomials?
Williams Math Lessons
Algebra 1 Polynomial Expressions and Factoring -110-
WARM UP Simplify. 1) (3x + 2y )+ 5x 2) (4h + 5j )− 3h 3) (−6a + 5b)+ 2a
KEY CONCEPTS AND VOCABULARY
A __________________________ is a real number, a variable, or the
product of real numbers and variables (Note: the variables must have
positive integer exponents to be a monomial).
The ___________________________________________________ is the sum
of the exponents of its variables.
A _________________________ is a monomial or a sum of monomials.
___________________________________________
means that the degrees of its monomial terms are
written in descending order.
The ___________________________________
_______________________ is the same as the degree
of the monomial with the greatest exponent.
EXAMPLES
EXAMPLE 1: IDENTIFYING POLYNOMIALS
Determine whether each expression is a polynomial. If it is a polynomial, classify the polynomial by the degree and number of terms. a) 2x 2 − 3x 3 + 4x b) 12x 2 + 10x −3 c) x
2 + 3x + 4x 2 d) 5
ADDING AND SUBTRACTING POLYNOMIALS MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the
operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
RATING LEARNING SCALE
4 I am able to
• add and subtract polynomials in real-world applications or in more challenging problems that I have never previously attempted
3 I am able to
• identify a polynomial and write polynomials in standard form • add and subtract polynomials
2 I am able to
• identify a polynomial and write polynomials in standard form with help • add and subtract polynomials with help
1 I am able to • identify the degree of a monomial
EXAMPLES OF MONOMIALS
EXAMPLES OF NOT
MONOMIALS 6 x
−3
p7
4y
4.75a2bc 3 9s 2t −5
g
8xyz
CLASSIFICATION OF POLYNOMIALS DEGREE NUMBER OF TERMS
0 Constant 1 Monomial
1 Linear 2 Binomial
2 Quadratic 3 Trinomial
3 Cubic 4 Polynomial with 4 terms
TARGET
Algebra 1 Polynomial Expressions and Factoring -111-
EXAMPLE 2: WRITING POLYNOMIALS IN STANDARD FORM
Write the polynomial in standard form. Then identify the leading coefficient. a) 4a2 − 7a + 3a5 b) 5h − 9 − 2h4 − 6h3 c) −2+ 6t 2 − 7t + 2t 2
EXAMPLE 3: ADDING POLYNOMIALS
Simplify. a) (2x 2 − 7 + 5x )+ (−4x 2 + 6x + 3) b) (5x + 7x 2 + 3)+ (−5x 2 + x 3 − 4)
EXAMPLE 4: SUBTRACTING POLYNOMIALS
Simplify. a) (3x + 2− x 2)− (4x − 5+ 2x 2) b) (12x 2 − 8x + 11)− (−14 + 10x 2 − 6x )
EXAMPLE 5: SIMPLIFYING USING GEOMETRIC FORMULAS
Express the perimeter as a polynomial. a) b)
EXAMPLE 6: ADDING AND SUBTRACTING POLYNOMIALS IN REAL-WORLD APPLICATIONS
The equation H = 3m + 120 and C = 4m + 84 represent the number of Miami Heat hats, H, and the number of Cleveland Cavalier hats, C, sold in m months at a sports store.
a) Write an equation for the total, T, of Heat and Cavalier hats sold.
b) Predict the number of Heat and Cavalier hats sold in 9 months.
RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)
Circle one: 4 3 2 1
Algebra 1 Polynomial Expressions and Factoring -112-
WARM UP Simplify.
1) x4 ⋅x 5 2) (3x 2yz )(−2xy 2z 3) 3)
5x 3y 2z 3
xyz
KEY CONCEPTS AND VOCABULARY
You can use the ___________________________________________ to multiply a monomial by a polynomial.
EXAMPLES
EXAMPLE 1: MULTIPLYING A POLYNOMIAL BY A MONOMIAL
Simplify. a) 2x 2(6x 2 − 2x + 5) b) −3x 2(x 2 + 3x − 8) EXAMPLE 2: SIMPLIFYING EXPRESSIONS WITH A PRODUCT OF A POLYNOMIAL AND A MONOMIAL
Simplify. a) 2x 2(−2x 2 + 5x )− 5(x 2 + 10) b) 3(5x 2 + x − 4)− x(4x 2 + 2x − 3)
MULTIPLYING A POLYNOMIAL BY A MONOMIAL MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the
operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
RATING LEARNING SCALE
4 I am able to
• multiply a polynomial by a monomial in more challenging problems that I have never previously attempted (such as solving equations)
3 I am able to
• multiply a polynomial by a monomial
2 I am able to • multiply a polynomial by a monomial with help
1 I am able to
• understand that the distributive property can be applied to polynomials
TARGET
Algebra 1 Polynomial Expressions and Factoring -113-
EXAMPLE 3: SIMPLIFYING USING GEOMETRIC FORMULAS
Express the area as a polynomial. a) b)
EXAMPLE 4: SOLVING EQUATIONS WITH POLYNOMIALS ON EACH SIDE
Solve. a) 2x(x + 4)+ 7 = (x + 9)+ x(2x + 1)+ 12
b) x(x 2 + 3x + 5)+ 2x 3 = 3x(x 2 + x + 5)+ 10
RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)
Circle one: 4 3 2 1
Algebra 1 Polynomial Expressions and Factoring -114-
WARM UP Simplify.
1) 9(x − 4) 2) −3(c + 6) 3)
14
(8y + 16)
KEY CONCEPTS AND VOCABULARY
METHODS FOR MULTIPLYING POLYNOMIALS
DISTRIBUTIVE PROPERTY METHOD FOIL METHOD
Example: (x + 4)(x − 3)
(x + 4)(x − 3)
x(x − 3)+ 4(x − 3)
x 2 − 3x + 4x − 12
x 2 + x − 12
Example: (x + 4)(x − 3)
(x + 4)(x − 3)
First Outer Inner Last
x ⋅x − 3 ⋅x 4 ⋅x 4 ⋅(−3)
x 2 + x − 12
EXAMPLES
EXAMPLE 1: FINDING THE PRODUCT OF TWO BINOMIALS USING THE DISTRIBUTIVE PROPERTY
Simplify using the distributive property. a) (x − 2)(x + 7) b) (2a + 7)(3a − 5) c) (r + 5)(5r + 10)
MULTIPLYING POLYNOMIALS MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the
operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
RATING LEARNING SCALE
4 I am able to
• multiply two binomials or a binomial by a trinomial in more challenging problems that I have never previously attempted
3 I am able to
• multiply two binomials or a binomial by a trinomial
2 I am able to • multiply two binomials or a binomial by a trinomial with help
1 I am able to
• understand the distributive property
TARGET
Algebra 1 Polynomial Expressions and Factoring -115-
EXAMPLE 2: FINDING THE PRODUCT OF TWO BINOMIALS USING THE FOIL METHOD
Simplify using the FOIL method. a) (x + 3)(x + 9) b) (5w − 2)(w + 3) c) (4k − 1)(3k − 7)
EXAMPLE 3: FINDING THE PRODUCT OF A BINOMIAL AND TRINOMIAL
Simplify using the distributive property. a) (2x − 6)(3x 2 + x − 1) b) (m − 1)(m3 − 4m + 12) c) (b
2 − 4b + 3)(b − 2)
EXAMPLE 4: SIMPLIFYING PRODUCTS
Simplify. a) (x + 2)[(x 2 + 3x − 6)+ (x 2 − 2x + 4)] b) [(x
2 + 3x − 7)− (x 2 − 2x + 6)](x − 4)
RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)
Circle one: 4 3 2 1
Algebra 1 Polynomial Expressions and Factoring -116-
WARM UP Simplify.
1) (z + 4)(z + 4) 2) (y − 3)(y − 3) 3) (q + 6)(q − 6)
KEY CONCEPTS AND VOCABULARY
EXAMPLES
EXAMPLE 1: SIMPLIFYING THE SQUARE OF A BINOMIAL (SUM)
Simplify. a) (x + 2)2 b) (5x + 2)2 c) (x
2 + 5)2
EXAMPLE 2: SIMPLIFYING THE SQUARE OF A BINOMIAL (DIFFERENCE)
Simplify. a) (x − 7)2 b) (2x − 1)2 c) (x
2 − 3)2
SPECIAL PRODUCTS MACC.912.A-APR.A.1: Understand that polynomials form a system analogous to the integers, namely, they are closed under the
operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
RATING LEARNING SCALE
4 I am able to
• simplify special products in more challenging problems that I have never previously attempted
3 I am able to
• find the square of a binomial • find the product of a sum and difference
2 I am able to
• find the square of a binomial with help • find the product of a sum and difference with help
1 I am able to
• understand that there are special rules to simplify the square of a binomial and the product of a sum and difference
MULTIPLYING SPECIAL CASES
THE SQUARE OF A BINOMIAL THE PRODUCT OF A SUM AND DIFFERENCE
(a + b)2 = (a + b)(a + b) = a2 + 2ab + b2 Or
(a – b)2 = (a – b)(a – b) = a2 – 2ab + b2
(a + b)(a – b) = a2 – b2
TARGET
Algebra 1 Polynomial Expressions and Factoring -117-
EXAMPLE 3: SIMPLIFYING THE PRODUCT OF A SUM AND DIFFERENCE
Simplify. a) (x + 3)(x − 3) b) (2x + 5)(2x − 5)
c) (x − 4)(4 + x ) d) (x2 + 6)(x 2 − 6)
EXAMPLE 4: SIMPLIFYING MORE CHALLENGING PROBLEMS WITH SPECIAL CASES
Simplify. a) (x + 2y )2 b) (a − 6b)(a + 6b)
c)
14
x + 2⎛⎝⎜
⎞⎠⎟
2
d) (x + 2)(x − 5)(x − 2)(x + 5)
e) (2x + 3)(2x − 3)(x + 1) f) [(4x + 1)(4x − 1)]+ [(x + 5)2]
RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)
Circle one: 4 3 2 1
Algebra 1 Polynomial Expressions and Factoring -118-
WARM UP Perform the indicated operation. 1) (x + 2x2 – 4) + (x2 – 3x + 9) 2) (x + 3)(x2 – 2) 3) (x – 1) – (5 + x)
KEY CONCEPTS AND VOCABULARY
EXAMPLES
EXAMPLE 1: FUNCTION OPERATIONS WITH LINEAR FUNCTIONS
Let f(x) = –2x + 6 and g(x) = 5x – 7. Use the functions f(x) and g(x) to find produce a new function h(x). a) h(x) = (f + g)(x) b) h(x) = (ƒ – g)(x) c) h(x) = (f • g)(x) d) h(x) = (f / g)(x)
FUNCTION OPERATIONS MACC.912.F-BF.A.1b: Combine standard function types using arithmetic operations.
RATING LEARNING SCALE
4 I am able to
• perform arithmetic operations with functions in more challenging problems that I have never previously attempted
3 I am able to
• perform arithmetic operations with functions
2 I am able to • perform arithmetic operations with functions with help
1 I am able to • understand that you can add, subtract, multiply, and divide functions
FUNCTION OPERATIONS
ADDITION (f + g)(x) = f(x) + g(x)
SUBTRACTION (f – g)(x) = f(x) – g(x)
MULTIPLICATION (f • g)(x) = f(x) • g(x)
DIVISION (f / g)(x) = f(x) / g(x), g(x) ≠ 0
TARGET
Algebra 1 Polynomial Expressions and Factoring -119-
EXAMPLE 2: FUNCTION OPERATIONS WITH LINEAR AND QUADRATIC FUNCTIONS
Let f (x ) = x 2 + 2x − 5 and g(x ) = x − 6 . Use the functions f(x) and g(x) to find produce a new function h(x). a) h(x) = (f + g)(x) b) h(x) = (ƒ – g)(x)
c) h(x) = (f • g)(x) d) h(x) = (f / g)(x)
EXAMPLE 3: FUNCTION OPERATIONS WITH LINEAR AND EXPONENTIAL FUNCTIONS
Let f (x ) = x − 1 and g(x ) = 3x + 4 . Use the functions f(x) and g(x) to find produce a new function h(x). a) h(x) = (f + g)(x) b) h(x) = (ƒ – g)(x)
c) h(x) = (f • g)(x) d) h(x) = (f / g)(x)
EXAMPLE 4: FUNCTION OPERATIONS WITH QUADRATIC AND EXPONENTIAL FUNCTIONS
Let f (x ) = −2x 2 + 4x − 8 and g(x ) = 2x − 5 . Use the functions f(x) and g(x) to find produce a new function h(x).
a) h(x) = (f + g)(x) b) h(x) = (ƒ – g)(x)
RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)
Circle one: 4 3 2 1
Algebra 1 Polynomial Expressions and Factoring -120-
WARM UP Multiply.
1) 3(x – 2) 2) x(x – 9) 3) (x + 5)(x – 9) 4) x2(x2 – 4x + 5)
KEY CONCEPTS AND VOCABULARY
You can work ______________________ to express a polynomial as the product of polynomials.
_________________________ – rewriting an expression as the product of polynomials. (un-distributing)
______________________________________________ – the largest quantity that is a factor of all the integers or
polynomials involved.
EXAMPLES
EXAMPLE 1: FINDING THE GREATEST COMMON FACTOR FROM A LIST OF INTEGERS
Find the greatest common factor of each list of numbers. a) 12 and 8 b) 7 and 20 c) 4, 12, and 26
EXAMPLE 2: FINDING THE GREATEST COMMON FACTOR FROM A LIST OF MONOMIALS
Find the greatest common factor of each list of monomials. a) x3 and x7 b) 6x5 and 4x3 c) 3xy2 and 12x2y
GREATEST COMMON FACTOR MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
RATING LEARNING SCALE
4 I am able to
• rewrite an expression as the product of the greatest common factor and the remaining polynomial in more challenging problems that I have never previously attempted
3 I am able to
• rewrite an expression as the product of the greatest common factor and the remaining polynomial
2 I am able to
• rewrite an expression as the product of the greatest common factor and the remaining polynomial with help
1 I am able to
• identify the greatest common factor
TARGET
Algebra 1 Polynomial Expressions and Factoring -121-
EXAMPLE 3: FACTORING THE GREATEST COMMON FACTOR
Factor the greatest common factor in each of the following polynomials. a) 15x2 + 100 b) 8m2 + 4m c) 3x2 + 6x d) 5x2 + 13y
e) 6x3 – 9x2 + 12x f) 14x3y + 7x2y – 7xy
g) 6(x + 2) – y(x + 2) h) xy(y + 1) – (y + 1)
EXAMPLE 4: SIMPLIFYING USING GEOMETRIC FORMULAS
Express the perimeter in factored form. a) b)
RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)
Circle one: 4 3 2 1
Algebra 1 Polynomial Expressions and Factoring -122-
WARM UP Factor out the greatest common factor.
1) x(x + 2) – 3(x + 2) 2) x2(x – 1) + (x – 1) 3) 4x(y + 12) + (y + 12)
KEY CONCEPTS AND VOCABULARY
___________________________________________ – factor a polynomial by grouping the terms of the polynomial
and looking for common factors.
FACTORING BY GROUPING (4 TERMS)
ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)
EXAMPLES
EXAMPLE 1: FACTORING A POLYNOMIAL BY GROUPING
Factor. a) x3 + 2x2 – 3x – 6 b) x3 + 4x + x2 + 4 c) 2x3 – x2 – 10x + 5
d) ab + 2a + 8b + 16 e) xy – 6x + 6y – 36 f) 9rs – 45r – 7s + 35
RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)
Circle one: 4 3 2 1
FACTORING BY GROUPING MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.A-SSE.A.1b: Interpret complicated expressions by viewing one or more of their parts as a single entity.
RATING LEARNING SCALE
4 I am able to
• factor polynomials by grouping in more challenging problems that I have never previously attempted
3 I am able to
• factor polynomials by grouping
2 I am able to • factor polynomials by grouping with help
1 I am able to
• understand that I can group polynomials to factor
TARGET
Algebra 1 Polynomial Expressions and Factoring -123-
WARM UP Multiply.
1) (x + 2)(x – 5) 2) (y – 7)(x – 1) 3) (x + y)(2x – y)
KEY CONCEPTS AND VOCABULARY
Steps for Factoring Trinomials with Leading Coefficient = 1 (x2 + bx + c) § Find two integers that multiply to c and add to b. § Write the binomial factors as (x + ___)(x + ___) filling in the blank with the two integers found § Check your answer by using the distributive property or the FOIL method
Example:
x 2 + 8x − 9 9 and − 1 are factors of − 9 that add to 8
(x + 9)(x − 1)
EXAMPLES
EXAMPLE 1: FACTORING TRINOMIALS IN THE FORM X2 + BX + C WHERE B AND C ARE POSITIVE
Factor. a) x
2 + 10x + 24 b) x2 + 7x + 12 c) x
2 + 17x + 42
EXAMPLE 2: FACTORING TRINOMIALS IN THE FORM X2 + BX + C WHERE B IS NEGATIVE AND C IS POSITIVE
Factor. a) x
2 − 14x + 33 b) x2 − 8x + 12 c) x
2 − 22x + 21
FACTORING X2 +BX + C MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
RATING LEARNING SCALE
4 I am able to
• factor trinomials of the form x2 + bx + c in more challenging problems that I have never previously attempted
3 I am able to
• factor trinomials of the form x2 + bx + c
2 I am able to • factor trinomials of the form x2 + bx + c with help
1 I am able to
• understand that some trinomials can be written as the product of two binomials
TARGET
Algebra 1 Polynomial Expressions and Factoring -124-
EXAMPLE 3: FACTORING TRINOMIALS IN THE FORM X2 + BX + C WHERE B IS POSITIVE AND C IS NEGATIVE
Factor. a) x
2 + 2x − 15 b) x2 + 13x − 48 c) x
2 + x − 20
EXAMPLE 4: FACTORING TRINOMIALS IN THE FORM X2 + BX + C WHERE B AND C ARE NEGATIVE
Factor. a) x
2 − 4x − 12 b) x2 − 2x − 24 c) x
2 − 7x − 18
EXAMPLE 5: FACTORING TRINOMIALS IN THE FORM X2 + BX + C AFTER FACTORING OUT A GCF
Factor. a) 2x 2 + 6x − 56 b) −x 2 + 6x − 5 c) x
2y + 10xy + 16y
EXAMPLE 6: APPLYING FACTORING TRINOMIALS TO GEOMETRIC FORMULAS
The area of a rectangle is given by the trinomial x2 + 12x + 20 .
a) What are the possible dimensions of the rectangle?
b) What are the exact dimensions if the width of the rectangle is 3 inches?
RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)
Circle one: 4 3 2 1
Algebra 1 Polynomial Expressions and Factoring -125-
WARM UP Write 2 different expressions that have a factor of (x + 6).
KEY CONCEPTS AND VOCABULARY
Steps for Factoring Trinomials with Leading Coefficient ≠ 1 (ax2 + bx + c) § Find two integers that multiply to ac and add to b. § Rewrite the trinomial by splitting the middle term (b term) into the two integers found. § Factoring by grouping § Check your answer by using the distributive property or the FOIL method
Example:
2x 2 − 13x − 24 ac = −48 and b = −13
2x 2 − 16x + 3x − 24 − 16 and 3 are factors of − 48 that add to − 13
2x(x − 8)+ 3(x − 8) Factor by grouping
(2x + 3)(x − 8)
EXAMPLES
EXAMPLE 1: FACTORING TRINOMIALS IN THE FORM AX2 + BX + C
Factor. a) 5x 2 − 13x + 6 b) 2x 2 + 9x − 5 c) 3x 2 + 23x − 36
RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)
Circle one: 4 3 2 1
FACTORING AX2 +BX + C MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
RATING LEARNING SCALE
4 I am able to
• factor trinomials of the form ax2 + bx + c in more challenging problems that I have never previously attempted
3 I am able to
• factor trinomials of the form ax2 + bx + c
2 I am able to • factor trinomials of the form ax2 + bx + c with help
1 I am able to
• understand that some trinomials can be written as the product of two binomials
TARGET
Algebra 1 Polynomial Expressions and Factoring -126-
WARM UP Multiply.
1) (2x – 7)(2x – 7) 2) (4x + 3)(4x – 3)
KEY CONCEPTS AND VOCABULARY
EXAMPLES
EXAMPLE 1: FACTORING PERFECT SQUARE TRINOMIALS
Factor. a) x
2 + 10x + 25 b) x2 − 18x + 81 c) 2x 2 − 24x + 72
EXAMPLE 2: FACTORING DIFFERENCE OF TWO SQUARES
Factor. a) x
2 − 16 b) x2 − 100 c) 12x 2 − 75
RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)
Circle one: 4 3 2 1
FACTORING SPECIAL CASES MACC.912.A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.
MACC.912.A-SSE.A.1a: Interpret parts of an expression, such as terms, factors, and coefficients.
RATING LEARNING SCALE
4 I am able to
• factor perfect-square trinomials and the differences of two squares in more challenging problems that I have never previously attempted
3 I am able to
• factor perfect-square trinomials and the differences of two squares
2 I am able to • factor perfect-square trinomials and the differences of two squares with help
1 I am able to
• understand that some polynomials can be written as the product of two binomials
FACTORING SPECIAL CASES
PERFECT SQUARE TRINOMIAL DIFFERENCE OF TWO SQUARES
a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2 Or
a2 – 2ab + b2 = (a – b)(a – b) = (a – b)2
a2 – b2 = (a + b)(a – b)
TARGET
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