1. x4 1. x(x3) 2. 3x2(x5) 3x7 3. 2(5x3) 10x3 4. x(6x2) 6x3 5. xy(7x2)
Warm Up
Multiply. x4
1. x(x3)
2. 3x2(x5)
3x7
3. 2(5x3)
10x3
4. x(6x2)
6x3
5. xy(7x2)
7x3y
6. 3y2(–3y)
–9y3

2. Objectives Multiply polynomials.
Use binomial expansion to expand binomial expressions that are raised to positive integer powers.

3. To multiply a polynomial by a monomial, use the Distributive Property and the Properties of Exponents.

4. Example 1: Multiplying a Monomial and a Polynomial
Find each product.
A. 4y2(y2 + 3)
4y2(y2 + 3)
4y2  y2 + 4y2  3
Distribute.
4y4 + 12y2
Multiply.
B. fg(f4 + 2f3g – 3f2g2 + fg3)
fg(f4 + 2f3g – 3f2g2 + fg3)
fg  f4 + fg  2f3g – fg  3f2g2 + fg  fg3
Distribute.
f5g + 2f4g2 – 3f3g3 + f2g4
Multiply.

5. Check It Out! Example 1
Find each product.
a. 3cd2(4c2d – 6cd + 14cd2)
3cd2(4c2d – 6cd + 14cd2)
3cd2  4c2d – 3cd2  6cd + 3cd2  14cd2
Distribute.
12c3d3 – 18c2d3 + 42c2d4
Multiply.
b. x2y(6y3 + y2 – 28y + 30)
x2y(6y3 + y2 – 28y + 30)
x2y  6y3 + x2y  y2 – x2y  28y + x2y  30
Distribute.
6x2y4 + x2y3 – 28x2y2 + 30x2y
Multiply.

6. To multiply any two polynomials, use the Distributive Property and multiply each term in the second polynomial by each term in the first.
Keep in mind that if one polynomial has m terms and the other has n terms, then the product has mn terms before it is simplified.

7. Example 2A: Multiplying Polynomials
Find the product.
(a – 3)(2 – 5a + a2)
Method 1 Multiply horizontally.
(a – 3)(a2 – 5a + 2)
Write polynomials in standard form.
Distribute a and then –3.
a(a2) + a(–5a) + a(2) – 3(a2) – 3(–5a) –3(2)
a3 – 5a2 + 2a – 3a2 + 15a – 6
Multiply. Add exponents.
a3 – 8a2 + 17a – 6
Combine like terms.

8. Example 2A: Multiplying Polynomials
Find the product.
(a – 3)(2 – 5a + a2)
Method 2 Multiply vertically.
a2 – 5a + 2
a – 3
Write each polynomial in standard form.
– 3a2 + 15a – 6
Multiply (a2 – 5a + 2) by –3.
a3 – 5a2 + 2a
Multiply (a2 – 5a + 2) by a, and align like terms.
a3 – 8a2 + 17a – 6
Combine like terms.

9. Example 2B: Multiplying Polynomials
Find the product.
(y2 – 7y + 5)(y2 – y – 3)
Multiply each term of one polynomial by each term of the other. Use a table to organize the products.
y –y –3 y2
–7y 5
The top left corner is the first term in the product. Combine terms along diagonals to get the middle terms. The bottom right corner is the last term in the product. y4
–y3
–3y2
–7y3
7y2
21y
5y2
–5y
–15
y4 + (–7y3 – y3 ) + (5y2 + 7y2 – 3y2) + (–5y + 21y) – 15
y4 – 8y3 + 9y2 + 16y – 15

10. Multiply horizontally. Write polynomials in standard form.
Check It Out! Example 2a
Find the product.
(3b – 2c)(3b2 – bc – 2c2)
Multiply horizontally.
Write polynomials in standard form.
(3b – 2c)(3b2 – 2c2 – bc)
Distribute 3b and then –2c.
3b(3b2) + 3b(–2c2) + 3b(–bc) – 2c(3b2) – 2c(–2c2) – 2c(–bc)
Multiply. Add exponents.
9b3 – 6bc2 – 3b2c – 6b2c + 4c3 + 2bc2
9b3 – 9b2c – 4bc2 + 4c3
Combine like terms.

11. x4 –4x3 x2 5x3 5x –2x2 8x –2 Check It Out! Example 2b
Find the product.
(x2 – 4x + 1)(x2 + 5x – 2)
Multiply each term of one polynomial by each term of the other. Use a table to organize the products.
x –4x x2 5x –2
The top left corner is the first term in the product. Combine terms along diagonals to get the middle terms. The bottom right corner is the last term in the product. x4
–4x3 x2
5x3
–20x2 5x
–2x2 8x –2
x4 + (–4x3 + 5x3) + (–2x2 – 20x2 + x2) + (8x + 5x) – 2
x4 + x3 – 21x2 + 13x – 2

12. 3. The number of items is modeled by
Lesson Quiz
Find each product.
1. 5jk(k – 2j)
5jk2 – 10j2k
2. (2a3 – a + 3)(a2 + 3a – 5)
2a5 + 6a4 – 11a3 + 14a – 15
3. The number of items is modeled by
0.3x x + 2, and the cost per item is modeled by g(x) = –0.1x2 – 0.3x + 5. Write a polynomial c(x) that can be used to model the total cost.
–0.03x4 – 0.1x x2 – 0.1x + 10
4. Find the product.
(y – 5)4
y4 – 20y y2 – 500y + 625
5. Expand the expression.
(3a – b)3
27a3 – 27a2b + 9ab2 – b3