Unit Essential Questions Are two algebraic expressions ...

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


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


WARM UP Simplify.

1) x4 ⋅x 5 2) (3x 2yz )(−2xy 2z 3) 3)

5x 3y 2z 3

xyz


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.


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



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


Circle one: 4 3 2 1


WARM UP Simplify.

1) 9(x − 4) 2) −3(c + 6) 3)

14

(8y + 16)


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



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


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)


Circle one: 4 3 2 1


WARM UP Simplify.

1) (z + 4)(z + 4) 2) (y − 3)(y − 3) 3) (q + 6)(q − 6)


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



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


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]


Circle one: 4 3 2 1


WARM UP Perform the indicated operation. 1) (x + 2x2 – 4) + (x2 – 3x + 9) 2) (x + 3)(x2 – 2) 3) (x – 1) – (5 + x)


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.


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


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)


Circle one: 4 3 2 1


WARM UP Multiply.

1) 3(x – 2) 2) x(x – 9) 3) (x + 5)(x – 9) 4) x2(x2 – 4x + 5)


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.


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


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)


Express the perimeter in factored form. a) b)


Circle one: 4 3 2 1


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)


___________________________________________ – 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


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.


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


WARM UP Multiply.

1) (x + 2)(x – 5) 2) (y – 7)(x – 1) 3) (x + y)(2x – y)


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.


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


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?


Circle one: 4 3 2 1


WARM UP Write 2 different expressions that have a factor of (x + 6).


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


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.



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


WARM UP Multiply.

1) (2x – 7)(2x – 7) 2) (4x + 3)(4x – 3)


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


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.



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

Unit Essential Questions Are two algebraic expressions ...

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