4.5 Multiplying Polynomials

Objective 1

Multiply a monomial and a polynomial.

Slide 4.5-3

To find the product of a monomial and a polynomial with more than one term we use the distributive property and multiplication of monomials.

Multiply a monomial and a polynomial.

As shown in Section 4.1, we find the product of two monomials by using the rules for exponents and the commutative and associative properties. For example

6 6 6 6 6 68 9 8 9 72 .m n m n m n

Do not confuse addition of terms with multiplication of terms. For instance,

but 5 5 57 2 9q q q 5 5 5 5 107 2 7 2 14 .q q q q

Slide 4.5-4

Solution:

Find the product.

4 22 3 2 5x x x

424 43 2 52 2 2x x xx x

6 5 46 4 10x x x

Slide 4.5-5

Multiplying Monomials and PolynomialsCLASSROOM EXAMPLE 1

Objective 2

Multiply two polynomials.

Slide 4.5-6

Multiply two polynomials.

We can use the distributive property repeatedly to find the product of any two polynomials. For example, to find the product of the polynomials x2 + 3x +5 and x − 4, think of x − 4 as a single quantity and use the distributive property as follows.

2 24 4 45 5 43 3x x xx x xx x

Now use the distributive property three more times to find x2(x − 4), 3x(x − 4), and 5(x − 4).

Multiplying PolynomialsTo multiply two polynomials, multiply each term of the second polynomial by each term of the first polynomial and add the products.

2 2 3 3 5 54 4 4x xx x x x x

3 2 24 3 12 5 20x x x x x 3 2 7 20x x x

Slide 4.5-7

Multiply (m3 − 2m + 1)·(2m2 + 4m + 3).

Solution:

3 2 3 3 2

2

2 4 3 2 2 2 4

2 3 1 2 1 4 1 3

m m m m m m m m m

m m m

5 4 3 3 2 22 4 3 4 8 6 2 4 3m m m m m m m m 5 4 3 22 4 6 2 3m m m m m

Slide 4.5-8

Multiplying Two PolynomialsCLASSROOM EXAMPLE 2

Multiply.

2(3 4 5)x x ( 4)x 212 16 20x x 3 23 4 5x x x

3 23 16 11 20x x x

Solution:

Slide 4.5-9

Multiplying Polynomials VerticallyCLASSROOM EXAMPLE 3

Find the product of

3 25 10 20x x 21 2

5 5x

3 22 4 8x x 5 4 22 4x x x

5 4 32 2 8x x x

Solution:

Slide 4.5-10

Multiplying Polynomials with Fractional Coefficients VerticallyCLASSROOM EXAMPLE 4

Objective 3

Multiply binomials by the FOIL method.

Slide 4.5-11

Multiply binomials by the FOIL method.

In algebra, many times the polynomials to be multiplied are binomials. For these products, the FOIL method reduces the rectangle method to a systematic approach without the rectangle.

Multiplying Binomials by the FOIL MethodStep 1: Multiply the two First terms of the binomials to get the

first term of the answer.

Step 2: Find the Outer product and Inner product and add them (when possible) to get the middle term of the answer.

Step 3: Multiply the two Last terms of the binomials to get the last term of the answer.

3 5x x

2F x L 15

O 5x I 3xSlide 4.5-12

Use the FOIL method to find the product.

2 6x x

2 6 2 8x x x 2 8 12x x

Solution:

2F x

O 6x I 2x

L 12

Slide 4.5-13

Using the FOIL MethodCLASSROOM EXAMPLE 5

Multiply 5 6 2 3 .x y

10 15 12 18xy x y

Solution:

5 6 2 3x y

F 10xy L 18

O 15x I 12y

Slide 4.5-14

Using the FOIL MethodCLASSROOM EXAMPLE 6

Find each product.

4 2 3y x y x

2 28 12 2 3y xy xy x Solution:

33 2 2 1x x x

2 28 14 3y xy x

3 23 2 1 4 2x x x x

3 23 2 3 2x x x 5 4 36 9 6x x x

Slide 4.5-15

Using the FOIL MethodCLASSROOM EXAMPLE 7