A polynomial is an expression with more than one term. For example (3x + 5) has two terms, 3x and 5. It is called a Binomial, meaning a two-term polynomial. Multiplying binomials is the focus of this unit.
Prior to starting this unit, please review the FOIL unit.
When multiplying binomials, a type of FOIL can be applied, though we will need to use it multiple times. It is generally referred to as the Distributive Property as well.
Example:
Multiply (x + 3)(x + 5)(x + 2)
Step 1: Use FOIL to multiply the first two binomials:
(x + 3)(x + 5) = x2 + 8x + 15
Step 2: Multiply the answer by the remaining binomial:
(x + 2)(x2 + 8x + 15)
x(x2 + 8x + 15) + 2(x2 + 8x + 15)
= x3 + 8x2 + 15x + 2x2 + 16x + 30
Step 3: Rearrange and combine like terms:
x3 + 8x2 + 2x2 + 15x + 16x + 30
= x3 + 10x2 + 31x + 30 * Answer
Example:
Multiply (x + 2)(x + 4)(x + 7)
Step 1: Use FOIL to multiply the first two binomials:
(x + 2)(x + 4) = x2 + 6x + 8
Step 2: Multiply the answer by the remaining binomial:
(x + 7)(x2 + 6x + 8)
x(x2 + 6x + 8) + 7(x2 + 6x + 8)
= x3 + 6x2 + 8x + 7x2 + 42x + 56
Step 3: Rearrange and combine like terms:
x3 + 6x2 + 7x2 + 8x + 42x + 56
= x3 + 13x2 + 50x + 56 * Answer
Multiply (x + 1)(x - 5)(x + 3)
Step 1: Use FOIL to multiply the first two binomials:
(x + 1)(x - 5) = x2 - 4x - 5
Step 2: Multiply the answer by the remaining binomial:
(x + 3)(x2 - 4x - 5)
x(x2 - 4x - 5) + 3(x2 - 4x - 5)
= x3 - 4x2 - 5x + 3x2 - 12x - 15
Step 3: Rearrange and combine like terms:
x3 - 4x2 + 3x2 - 5x - 12x - 15
= x3 - x2 - 17x - 15 * Answer
Multiply (2x + 1)(x + 4)(x - 2)
Step 1: Use FOIL to multiply the first two binomials:
(2x + 1)(x + 4) = 2x2 + 9x + 4
Step 2: Multiply the answer by the remaining binomial:
(x - 2)(2x2 + 9x + 4)
x(2x2 + 9x + 4) + -2(2x2 + 9x + 4)
= 2x3 + 9x2 + 4x + -4x2 + -18x + - 8
Step 3: Rearrange and combine like terms:
2x3 + 9x2 - 4x2 + 4x - 18x - 8
= 2x3 + 5x2 - 14x - 8 * Answer
Now you try:
Multiply (x + 1)(x + 8)(x + 2)
Enter your answer in the space provided. Use the caret key (^) to enter the exponents.
Now you try:
Multiply (x + 3)(x + 1)(x + 7)
Enter your answer in the space provided. Use the caret key (^) to enter the exponents.