Download Multiplying Polynomials

Int. Alg. Notes
Section 5.2
Page 1 of 6
Section 5.2: Multiplying Polynomials
Big Idea: Polynomials are the most important topic in algebra because any equation that can be written using
addition, subtraction, multiplication, division, integer powers, or roots (which are rational powers) can be
solved by converting the equation into a polynomial equation. The second step toward acquiring this awesome
power is to be able to multiply polynomials using the distributive property (or other tricks).
Big Skill: You should be able to multiply polynomials using the distributive property and some special product
formulas.
Skill #1: Multiplying Monomials
When multiplying monomials:
 Multiply the coefficients to get a single new coefficient.
 Add exponents of common variables to get simplified variable factors.
Example:
 2a b  6a b    2  6     a
3
 

 a2    b  b4 
 aaaaa   bbbbb 
2 4
3
  12   a 3 2  b14 
 12a 5b5
Practice:
1. Multiply (3xy2)(-5x2y3)
 2   15 
2. Multiply  x 4   x 
 3  8 
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.2
Page 2 of 6
Skill #2: Multiplying a Monomial and a Polynomial
When multiplying a monomial and a polynomial, use the extended form of the distributive property:
Extended form of the Distributive Property:
a  b1  b2  b3   bn   ab1  ab2  ab3   abn
Example:
2 x 2  x 2  3x  5  2 x 2  x 2  2 x 2  3x  2 x 2   5
 2 x 4  6 x3  10 x 2
Practice:
1. Multiply 
1 34 2
1
yz  yz  8 y  
2
4
3
Skill #3: Multiplying a Binomial and a Binomial
When multiplying a binomial and a binomial, you can use one of four techniques:
1. Distributive Property Technique
Example:
 3x  4  2 x  9    3x  4   2 x   3x  4    9 
 3 x  2 x  4  2 x  3 x   9   4   9 
 6 x 2  8 x  27 x  36
 6 x 2  19 x  36
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.2
Page 3 of 6
2. Vertical Multiplication Technique
Example:
 3x  4  2 x  9  
3x  4
 2x  9
 27 x  36
6x
2

2
 19 x  36
6x
8x
3. Table Multiplication Technique (not in book)
Example:
To Calculate  3x  4 2 x  9  :
To Calculate 3429:

20
9
30
600
270
4
80
36
= 600 + 80 + 270 + 36
= 986

2x
-9
3x
6x2
-27x
4
8x
-36
= 6x2 + 8x – 27x – 36
= 6x2 – 19x – 36
4. FOIL Technique (ONLY works for binomials)
FOIL  First, Outside, Inside, Last
Practice:
1. Multiply  2a  5 3a  4
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.2
Page 4 of 6
2. Multiply  2 x  5 4 x  7 
Skill #4: Multiplying a Polynomial and a Polynomial
When multiplying a binomial and a binomial, you can use one of two techniques:
1. Distributive Property Technique
Example:
 2 x  3  x 2  5 x  2    2 x  3   2 x  3   5 x    2 x  3   2 
 2 x  x 2  3  x 2  2 x  5 x  3  5 x  2 x   2   3    2 
 2 x3  3x 2  10 x 2  15 x  4 x  6
 2 x3  13x 2  11x  6
2. Table Multiplication Technique (not in book)
Example:
x2
5x
-2

3
2
2x
2x
10x
-4x
3
3x2
15x
-6
Notice that like terms line up on the diagonals…
Practice:
1. Multiply  x 2  4 x  2  2 x 2  x  5 
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.2
Page 5 of 6
Skill #5: Recognizing Special Binomial Products
When multiplying a pair of “conjugate” binomials, or when multiplying a binomial with itself, you can use the
following formulas:
1. Product of conjugate binomials
Warm-up exercise: Multiply the following pairs of conjugate binomials:
(x + 7)(x – 7) =
(3x + 12)(3x – 12) =
(m + n)(m – n) =
Notice the pattern: the middle terms always add to zero, and you are left with the square of the first term
minus the square of the second term. Verbally, we can say that the product of conjugate binomials is the
square of the first term minus the square of the second term. Mathematically, we describe this pattern
as:
The Product of Conjugate Binomials, or, a Difference of Two Squares
(A + B)(A – B) = A2 – B2
2. Square of a binomial
Warm-up exercise: Square the following binomials:
(x + 3)2 = (x + 3) (x + 3) =
(3x + 12)2 =
(m + n)2 =
Notice the pattern:

The answer is a trinomial.

The first term of the trinomial is the square of the first term of the binomial.

The second term of the trinomial is twice the product of the first and last terms of the binomial.

The last term of the trinomial is the square of the last term of the binomial.
This pattern is described mathematically as:
The Square of a Binomial, or, a Perfect Square Trinomial
(A + B)2 = A2 + 2AB + B2
(A – B)2 = A2 – 2AB + B2
Practice:
1. Multiply  2 x  5 2 x  5
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 5.2
Page 6 of 6
2. Multiply  4 y  z  4 y  z 
3. Multiply  n  8 
2
4. Multiply  7 z  2 
2
5. Multiply  3x  8 y 2 
2
If f and g are two functions, then

The new function that can be made by multiplying them together is called f  g: (f  g)(x) = f (x)  g(x).
Practice:
1. If f  x   2x  9 and g  x   3x2  4 x  7 ; compute (f  g)(x), (f  g)(2), and g(x + 3).
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.