Download 3.2 Multiplying Polynomials

3-2
Multiplying Polynomials
Objectives
Multiply polynomials.
Use binomial expansion to expand
binomial expressions that are raised to
positive integer powers.
Holt McDougal Algebra 2
3-2
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
3-2
Multiplying Polynomials
Example 1: 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)
Multiply. Add exponents.
Combine like terms.
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Example 2: Multiplying Polynomials
Find the product.
(y2 – 7y + 5)(y2 – y – 3)
y2
y2
y4
–y
–3
–y3 –3y2
–7y –7y3
7y2
5
–5y –15
5y2
1. Multiply row and column
2. Add terms along diagonals
21y
y4 + (–7y3 – y3 ) + (5y2 + 7y2 – 3y2) + (–5y + 21y) – 15
y4 – 8y3 + 9y2 + 16y – 15
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
You DO:
(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.
Combine like terms.
9b3 – 6bc2 – 3b2c – 6b2c + 4c3 + 2bc2
9b3 – 9b2c – 4bc2 + 4c3
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Example 3
Find the product.
(2x – 1)3
(2x – 1)(2x – 1)(2x – 1)
Write in expanded form.
Multiply the last two
binomial factors.
Distribute 2x and then –1.
2x(4x2) + 2x(–4x) + 2x(1) – 1(4x2) – 1(–4x) – 1(1)
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Notice the coefficients of the variables in the final
product of (a + b)3. these coefficients are the numbers
from the third row of Pascal's triangle.
Each row of Pascal’s triangle gives the coefficients of the
corresponding binomial expansion. The pattern in the table
can be extended to apply to the expansion of any binomial
of the form (a + b)n, where n is a whole number.
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
This information is formalized by the Binomial
Theorem, which you will study further in Chapter 11.
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Check It Out! Example 5
Expand each expression.
a. (x + 2)3
Identify the coefficients for n = 3, or row 3.
1331
[1(x)3(2)0] + [3(x)2(2)1] + [3(x)1(2)2] + [1(x)0(2)3]
x3 + 6x2 + 12x + 8
b. (x – 4)5
1 5 10 10 5 1
Identify the coefficients for n = 5, or row 5.
[1(x)5(–4)0] + [5(x)4(–4)1] + [10(x)3(–4)2] + [10(x)2(–4)3]
+ [5(x)1(–4)4] + [1(x)0(–4)5]
x5 – 20x4 + 160x3 – 640x2 + 1280x – 1024
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
Check It Out! Example 5
Expand the expression.
c. (3x + 1)4
14641
Identify the coefficients for n = 4, or row 4.
[1(3x)4(1)0] + [4(3x)3(1)1] + [6(3x)2(1)2] + [4(3x)1(1)3]
+ [1(3x)0(1)4]
81x4 + 108x3 + 54x2 + 12x + 1
Holt McDougal Algebra 2
3-2
Multiplying Polynomials
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
0.3x2 + 0.1x + 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 – 0.1x3 + 1.27x2 – 0.1x + 10
4. Find the product.
(y – 5)4 y4 – 20y3 + 150y2 – 500y + 625
5. Expand the expression.
(3a – b)3 27a3 – 27a2b + 9ab2 – b3
Holt McDougal Algebra 2