3.2B Multiplying Polynomials
Objectives:
A.APR.5: Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and
y for a positive integer n, where x and y are any numbers, with coefficients
determined for example by Pascal’s Triangle.
A.APR.1: Understand that polynomials form a system analogous to the integers, namely, they
are closed under the operations of addition, subtraction, and multiplication; add,
subtract, and multiply polynomials.
For the board: You will be able to multiply polynomials.
You will be able to use binomial expansion to expand binomial expressions that are
raised to positive integer powers.
Anticipatory Set:
Recall:
(a + b)2 = (a + b)(a + b)
(a + b)3 = (a + b)(a + b)(a + b)
(a + b)4 = (a + b)(a + b)(a + b)(a + b)
This multiplication is called binomial expansion.
Instruction:
Binomial Expansion
(a + b)0 = 1
(a + b)1 = a + b
(a + b)2 = a2 + 2ab + b2
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a + b)4 = a4 + 4a3b + 6a2b2 + 3ab3 + b4
(a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5
Pascal’s Triangle
(Coefficients)
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Binomial Expansion
For a binomial expansion of the form (a + b)n, the following statements are true.
1. There are n + 1 terms.
2. The coefficients are the numbers from the nth row of Pascal’s triangle.
The triangle begins with a row zero.
3. The exponent of “a” is “n” in the first term, and decreases by 1 in each successive term.
4. The exponent of “b” is 0 in the first term, and increases by 1 in each successive term.
5. The sum of the exponents in any term is n.
Open the book to page 161 and read example 5.
Example: Expand each expression.
Steps:
1. Write the coefficients from Pascal’s Triangle well spaced with plus signs in
front of them.
2. Next write the first term to the appropriate power. (n down to 0)
3. Next write the second term to the appropriate power. (0 up to n)
4. Simplify.
a. (k – 5)4
k4 + 4k3(-5) + 6k2(-5)2 + 4k(-5)3 + 1(-5)4
k4 – 20k3 + 150k2 – 500k + 625
b. (4m – 3)3
(4m)3 + 3(4m)2(-3) + 3(4m)(-3)2 + (-3)3
64m3 – 144m2 + 108m - 27
White Board Activity:
Practice: Expand each expression.
a. (x + 2)3
x3 + 3x2(2) + 3x(2)2 + 23 = x3 + 6x2 + 12x + 8
b. (x – 4)5
x5 + 5x4(-4) + 10x3(-4)2 + 10x2(-4)3 + 5x(-4)4 + (-4)5
x5 – 20x4 + 160x3 – 640x2 + 1280x - 1024
c. (3x + 1)4
(3x)4 + 4(3x)3(1) + 6(3x)2(1)2 + 4(3x)(1)3 + 14
81x4 108x3 + 54x2 + 12x + 1
Assessment:
Question student pairs.
Independent Practice:
Text: pg. 162 - 163 prob. 10 – 17, 27 – 34, 41, 44, 47, 50.
For a Grade:
Text: pg. 162 - 163 prob. 32, 34, 50.