Transcript Lesson 3.2A

Multiplying Polynomials
• How do we multiply polynomials?
•How do we use binomial expansion to
expand binomial expressions that are
raised to positive integer powers?
Holt McDougal Algebra 2
Multiplying Polynomials
To multiply a polynomial by a monomial, use
the Distributive Property and the Properties
of Exponents.
Holt McDougal Algebra 2
Multiplying Polynomials
Example 1: Multiplying a Monomial and a Polynomial
Find each product.


A. 4 y 2 y 2  3
Distribute.
4y4  12 y 2

B. fg f 4  2 f 3 g  3 f 2 g 2  fg 3

f 5g  2 f 4 g 2  3 f 3g 3  f 2 g 4
Holt McDougal Algebra 2
Distribute.
Multiplying Polynomials
Example 1: Multiplying a Monomial and a Polynomial
Find each product.

C. 3cd 2 4c 2 d  6cd  14cd 2
1 2 c 3d 3  1 8 c 2d 3  4 2 c 2d


4

D. x 2 y 6 y 3  y 2  28 y  30
6x2y4 x2y3  28x2 y 2  30 x2 y
Holt McDougal Algebra 2
Distribute.
Distribute.
Multiplying Polynomials
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.
Holt McDougal Algebra 2
Multiplying Polynomials
Example 2A: Multiplying Polynomials
Find the product.
a  32  5a  a 2 
Method 1 Multiply horizontally.
Write polynomials in standard form.
a  3 a 2  5a  2
Distribute a and then –3.



a(a2) + a(–5a) + a(2) – 3(a2) – 3(–5a) –3(2)
a3  5 a 2  2 a  3 a 2  1 5 a  6
Multiply. Add exponents.
a  8a  17 a  6
Combine like terms.
3
2
Holt McDougal Algebra 2
Multiplying Polynomials
Example 2B: Multiplying Polynomials
Find the product.
a  32  5a  a 2 
Method 2 Multiply vertically.
a 2  5a  2
a 3
 3a  1 5 a  6
a3  5 a 2  2 a
2
3
a  8a  17 a  6
2
Holt McDougal Algebra 2
Write each polynomial in
standard form.
Multiply (a2 – 5a + 2) by –3.
Multiply (a2 – 5a + 2) by a, and
align like terms.
Combine like terms.
Multiplying Polynomials
Example 3: Multiplying Polynomials
Find the product.
y
2


 7 y  5 y2  y  3
Multiply each term of one polynomial by each term of the other. Use
a table to organize the products.
y2
y2
4
y
–y
y
–3
3
–7y  7 y 3 7 y 2
5
 3y
2
21y
5y2  5 y  1 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.
y  8y  9 y  16 y  15
4
3
Holt McDougal Algebra 2
2
Multiplying Polynomials
Example 4: Multiplying Polynomials
Find the product.
3b  2c  bc  2c 2  3b2 
Method 1 Multiply horizontally.
3b  2c 3b 2  2c 2  bc 
Write polynomials in standard form.
Distribute 3b and then –2c.
3b(3b2) + 3b(–2c2) + 3b(–bc) – 2c(3b2) – 2c(–2c2) – 2c(–bc)
9b  6 b c  3 b c  6 b c  4 c  2 b c
3
2
2
9b  9 b c  4 b c  4 c
3
2
Holt McDougal Algebra 2
3
2
2
3
2
Multiply.
Add exponents.
Combine like terms.
Multiplying Polynomials
Example 5: Multiplying Polynomials
Find the product.
x
2

 4 x  1 x  5x  2
2

Multiply each term of one polynomial by each term of the other. Use
a table to organize the products.
–4x
x2
x2
x4  4 x
5x
–2
 2x
3
x
2
 2 0 x 2 5x
3
5x
1
2
8x
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  x 3  2 1 x 2  1 3 x  2
Holt McDougal Algebra 2
Multiplying Polynomials
Lesson 3.2 Practice A
Holt McDougal Algebra 2