Math 101 Lecture Notes Ch. 4.3 4.3 Multiplying Polynomials
To multiply two monomials, we will use the rules of exponents and the commutative
property of multiplication. We have already practiced this skill in section 4.1. Let's
review what we did.
Demonstration Problems
1. (a) Simplify
Practice Problems
1. (b) Simplify
2x3 • 3x4
5x2 • 7x6
2. (a) Simplify
2. (b) Simplify
(–2ab2)( –3a2b4)
(–4xy3)( –7x2y)
Answers: 1. (b) 35x8; 2. (b) 28x3y4
To multiply a monomial by any polynomial of 2 or more terms, we will first apply the
distributive property, then the commutative property and the rules of exponents.
For example:
2x2(2x3 + x2 + 5) =
=
2x2 (2x3 + x2 + 5) Apply the distributive property. 2x2 • 2x3 + 2x2 • x2 + 2x2 • 5 Apply the commutative property of multiplication. = 2 • 2 • x2 • x3 + 2 • x2 • x2 + 2 • 5 • x2
= 4x5 + 2x4 + 10x2
Demonstration Problems
3. (a) Multiply
Apply the rules of exponents and simplify the result. Practice Problems
3. (b) Multiply
2x (3x + 2)
5a (a + 1)
4. (a) Multiply
4. (b) Multiply
2
2
5x (3x + 2x – 1)
3m2 (2m2 + 5m – 4)
Answers: 3. (b) 5a2 + 5a; 4. (b) 6m4 + 15m3 – 12m2
Page 1 of 5 Math 101 Lecture Notes Ch. 4.3 To multiply a binomial by a polynomial of 2 or more terms, we apply the distributive
property twice and then simplify the result.
For example:
(x + 2)(x2 + 3x + 4) = (x + 2)(x2 + 3x + 4)
Apply the distributive property. = (x + 2)x2 + (x + 2)3x + (x + 2)4
Apply the distributive property again. = x • x2 + 2x2 + x • 3x + 2 • 3x + x • 4 + 2 • 4
Simplify the result. = x3 + 2x2 + 3x2 + 6x + 4x + 8
= x3 + 5x2 + 10x + 8
5. (a)
Demonstration Problems
Multiply
(3x – 2)(2x + 3)
6. (a)
Multiply
(x + 4)(2x2 + 3x + 5)
Practice Problems
5. (b) Multiply
(2x – 1)(3x + 1)
6. (b) Multiply
(x + 3)(5x2 + 2x + 6)
Answers: 5. (b) 6x2 – x – 1; 6. (b) 5x3 + 17x2 + 12x + 18
Applying the distributive property twice can be cumbersome. Several more efficient
methods of multiplying polynomials follow.
Page 2 of 5 Math 101 Lecture Notes Ch. 4.3 Vertical Stacking Method
Multiply polynomials as you would multiply whole numbers applying the rules for
multiplying place values to the polynomial terms.
Multiply and simplify (2a – 5b)(3a – 2b).
Demonstration Problems
7. (a) Multiply and simplify
2a – 5b 3a – 2b –4ab + 10b2 6a2–15ab .... 6a2–19ab + 10b2 Practice Problems
7. (b) Multiply and simplify
(5x + 2)(x – 3)
(x + 5)(2x – 1)
Answer: 7. (b) 2x2 + 9x – 5
FOIL Method
for Multiplying Binomials
Last First F O I L (3x – 4)(2x + 3) = 6x2 + 9x – 8x – 12 = 6x2 + x – 12 Inner Outer Demonstration Problems
8. (a) Multiply and simplify
(3x – 2)(x – 2)
Practice Problems
8. (b) Multiply and simplify
(x + 1)(3x – 4)
Answer: 8. (b) 3x2 – x – 4
Page 3 of 5 Math 101 Lecture Notes Ch. 4.3 Geometric Area Model (Box Method)
Recall that the area of a rectangle is the
product of its base and its height.
5 We can use monomials as the base and
height as well:
5x Area = 3x 3 3 • 5 = 15 Area = 3x • 5x = 15x2 We can also “stack” several rectangles together into a large rectangle with segments of
the outer dimensions labeled with monomials:
5x 4 The total area of the rectangle is
Area = 15x2 Area = 3x • 4 = 12x Area = Area = 3x 3x • 5x = 15x2 + 12x + 10x + 8 = 15x2 + 22x + 8
2 2 • 5x = 10x 2 • 4 = 8 The geometric model can be used to simplify (3x – 4)(2x + 3) as follows:
(3x – 4)(2x + 3)
2x 2x 3 3x 6x2 9x –4 –8x –12 3 3x ⇒
–4 Therefore, (3x – 4)(2x + 3) = 6x2 + 9x – 8x – 12 = 6x2 + x – 12
Demonstration Problems
9. (a) Multiply
(2x + 1)(5x + 1)
Practice Problems
9. (b) Multiply
(6a + 2)(a + 2)
10. (a) Multiply
10. (b) Multiply
(x – 2)(3x – 1)
(x – 5)(4x – 3)
Answers: 9. (b) 6a2 + 14a + 4;
Page 4 of 5 10. (b) 4x2 – 23m + 15
Math 101 Lecture Notes Ch. 4.3 To multiply a binomial by a trinomial, we can expand the "box" into a rectangle of 3
columns and 2 rows as follows:
x2
(x + 2)(x2 + 3x – 5)
⇒
3x
–5
x
2
Demonstration Problems
11. (a) Multiply
(5x +1)(3x2 + 2x – 1)
Practice Problems
11. (b) Multiply
(3m + 1)(2m2 + 5m – 4)
Answer: 11. (b) 6m3 + 17m2 – 7m – 4
Page 5 of 5