Polynomials Lesson 3Multiplying and Factoring Polynomials using the Distributive Method

Todays Objectives

Students will be able to demonstrate an understanding of the multiplication of polynomial expressions, including: Generalize and explain a strategy for multiplication of

polynomials Identify and explain errors in a solution for a

polynomial expression Generalize and explain strategies used to factor a

trinomial Express a polynomial as a product of its factors

Multiplying Polynomials

Today we will look at a strategy for multiplying and factoring polynomials: Distributive Method

When binomials have negative terms, it is difficult to show their product using algebra tiles, but we can use a rectangle diagram or the distributive property.

Algebra tiles are limited to multiplying 2 binomials. The distributive property can be used to multiply any polynomials

Distributive Property Consider the following example:

First we multiply each term in the first brackets by both terms in the second bracket. We call this “distributing”

Next, we combine like terms Like terms are terms that have the same variables which have

the same degree, or exponent In this case -8b and -3b are like terms. Both terms have the

variable “b”, with the same degree, or exponent (1). b2 is not a like term because “b” has an exponent, or degree of 2

Example (You do)

Use the distributive property to find the product of the binomials

Solution:

Notice from the trinomial:

8 is the sum of 3 and 5

15 is the product of 3 and 5

The leading coefficient of the trinomial is 1, as are the leading coefficients of the binomial factors

Example

Factor the trinomial x2 – 4x – 21 into two binomials.

Answer will be in the form (x + a)(x + b), where we know the following:

a + b = -4

(a)(b) = -21

So we need to find two numbers that ADD to -4, and MULTIPLY to -21

Example

Find all factors of -21

(1)(-21), (-1)(21), (3)(-7), (-3)(7)

Find one set of factors that add to -4

(3)(-7)

So, a = 3, b = -7 (x + 3)(x – 7)

Example (You do)

Factor the trinomial z2 – 12z + 35

Solution: Factors of 35 are (1)(35), (-1)(-35), (5)(7), (-5)(-7)

Two factors that add to -12 are (-5)(-7)

So, a = -5, b = -7

(z – 5)(z – 7)

Example List all the factors for the polynomial 2x2 – 10x + 12

Solution: First, remove the GCF 2(x2 – 5x + 6)

*you must include the factored out 2 as a factor in your final answer!

Now, factor the trinomial: (x + a)(x + b), a + b = -5, (a)(b) = +6

Find all the factors of +6 (1)(6), (2)(3), (-1)(-6), (-2)(-3)

Find one set of factors that add to – 5 (-2)(-3)

So, a = -2, b = -3 Factors of the polynomial 2x2 – 10x + 12 are:

(2)(x - 2)(x – 3)

Example (You do)

Factor -4t2 – 16t + 128

Solution:

Remove the GCF (-4):

Find two factors of -32 that add to +4: (-4)(8)

a = -4, b = 8

Distributive Method

The distributive property can be used to perform any polynomial multiplication. Each term of one polynomial must be multiplied by each term of the other polynomial.

Using the distributive property to multiply two polynomials

Multiply (2h + 5)(h2 + 3h – 4)

Solution: Multiply each term in the trinomial by each term in the binomial. Write the terms in a list.

Example

-3f2(4f2 – f – 6): -12f4 + 3f3 + 18f2

3f(4f2 – f – 6): 12f3 – 3f2 – 18f

-2(4f2 – f – 6): -8f2 + 2f + 12

Add: -12f4 +15f3 + 7f2 –16f + 12

(-3f2 + 3f – 2)(4f2 – f – 6)Use the distributive property. Multiply each term in the 1st trinomial by each term in the 2nd trinomial. Align like terms.

Check your answers!

One way to check that your product is correct is to substitute a number for the variable(s) in both your answer and the original question. If both sides are equal, then your answer is correct. For this example, let’s let f = 1.

= (-3f2 + 3f – 2)(4f2 – f – 6) = -12f4 + 15f3 + 7f2 – 16f + 12

= [-3(1)2 + 3(1) – 2][4(1)2 – 1 – 6] = -12(1)4 + 15(1)3 + 7(1)2 – 16(1) + 12

= (-2)(-3) = 6

= 6 = 6

Since the left side equals the right side, the product is likely correct.

Example (You do)

Use the distributive property to multiply (3x – 2y)(4x – 3y + 5)

Solution:

= 3x(4x – 3y – 5) – 2y(4x – 3y + 5)

= 3x(4x) + 3x(-3y) + 3x(-5) – 2y(4x) – 2y(-3y) – 2y(5)

= 12x2 – 9xy – 15x – 8xy + 6y2 – 10y

= 12x2 – 9xy – 8xy – 15x + 6y2 – 10y collect like terms

= 12x2 – 17xy – 15x + 6y2 – 10y

Example

Simpify (2c – 3)(c + 5) + 3(c – 3)(-3c + 1)

Solution:

Use the order of operations. Multiply before adding and subtracting. Then, combine like terms.

= 2c(c + 5) – 3(c + 5) + 3[c(-3c + 1) – 3(-3c + 1)]

= 2c2 + 10c – 3c – 15 + 3[-3c2 + c + 9c – 3]

= 2c2 + 7c – 15 + 3[-3c2 + 10c – 3]

= 2c2 + 7c – 15 – 9c2 + 30c – 9

= -7c2 + 37c – 24

Example (You do)

Simplify

Solution:

Homework

pg. 186-187, # 5,7,9,11,15-17, 21