4.5 Multiplying Polynomials
Learning Objectives
After completing this section, you should be able to:
- Multiply a monomial by a monomial.
- Multiply a polynomial by a monomial.
- Multiply a binomial by a binomial using the Distributive Property (FOIL).
- Multiply a binomial by a trinomial or larger polynomial.
- Simplify and write the product in standard form.
Multiplying a Monomial by a Monomial
When multiplying two monomials, multiply the coefficients and add the exponents of like variables.
Rule:
(aᵐ)(aⁿ) = aᵐ⁺ⁿ
Example 4.5.1
Simplify (3x²)(4x³)
Solution
(3x²)(4x³) = 12x⁵
Example 4.5.2
Simplify (–2y⁴)(5y³)
Solution
(–2)(5)y⁷ = –10y⁷
Example 4.5.3
Simplify (4p³q²)(–3p²q)
Solution
(4)(–3)p⁵q³ = –12p⁵q³
Try It
Multiplying a Monomial by a Polynomial
Use the Distributive Property: multiply the monomial by each term of the polynomial.
Rule:
a(b + c + d) = ab + ac + ad
Example 4.5.4
Simplify 3x(2x² + 5x – 1)
Solution
6x³ + 15x² – 3x
Example 4.5.5
Simplify –2y²(4y³ – 3y + 6)
Solution
–8y⁵ + 6y³ – 12y²
Example 4.5.6
Simplify –2a²b(5ab – 3b² + 2)
Solution
–10a³b² + 6a²b³ – 4a²b
Try It
Multiplying Two Binomials (FOIL)
When multiplying two binomials, use the Distributive Property twice.
A common shortcut is the FOIL method: First, Outer, Inner, Last.
Example 4.5.7
Simplify (x + 3)(x + 5)
Solution
x² + 8x + 15
Example 4.5.8
Simplify (2y – 4)(y + 6)
Solution
2y² + 8y – 24
Example 4.5.9
Simplify (x – 4)(x – 4)
Solution
x² – 8x + 16
Try It
Multiplying Larger Polynomials
When one or both polynomials have more than two terms, apply the Distributive Property systematically.
Example 4.5.10
Simplify (x² + 2x + 3)(x + 4)
Solution
x³ + 6x² + 11x + 12
Example 4.5.11
Simplify (y – 3)(2y² – y + 5)
Solution
2y³ – 7y² + 8y – 15
Example 4.5.12
Simplify (x³ + 2x² + x + 1)(x – 1)
Solution
x⁴ + x³ – x² – 1
Try It
Key Takeaways
- Multiply coefficients and add exponents of like bases.
- Use the Distributive Property to expand products.
- The FOIL method is a shortcut for binomials.
- Write final answers in standard form.
