Pengi Editor's Note: This article was originally published by Think Academy. We're sharing it here for educational value. Think Academy is a leading K-12 math education provider.
How to Multiply Polynomials: From 1 Term to 3 Terms
What are Polynomials?
A polynomial is a mathematical expression made by adding, subtracting, or multiplying numbers and variables, with exponents that are whole numbers.
Example: 3𝑥² + 2𝑥 − 5 has three terms: 3𝑥², 2𝑥, and -5.
Polynomial Multiplication Explained
Multiplication of Polynomials means multiplying each term in the first polynomial by each term in the second one.
1 Term × 1 Term (Monomial × Monomial)
- Multiply the numbers
- Multiply the variables by adding exponents
Example: 3𝑥 × 4𝑥² = 3 × 4 × 𝑥¹⁺² = 12𝑥³
1 Term × 2 Terms (Monomial × Binomial)
Multiply the single term by each term in the binomial.
Example: 2𝑥 × (𝑥 + 5) = 2𝑥² + 10𝑥
2 Terms × 2 Terms (Binomial × Binomial)
Use the FOIL method: First, Outer, Inner, Last.
Example: (𝑥 + 3) × (𝑦 + 2) = 𝑥𝑦 + 2𝑥 + 3𝑦 + 6
2 Terms × 3 Terms (Binomial × Trinomial)
Multiply each term in the first polynomial by each term in the second.
Example: (𝑥 + 4) × (𝑥² + 2𝑥 + 1)
= 𝑥 × (𝑥² + 2𝑥 + 1) + 4 × (𝑥² + 2𝑥 + 1)
= 𝑥³ + 2𝑥² + 𝑥 + 4𝑥² + 8𝑥 + 4
Always remember to combine like terms!
= 𝑥³ + 6𝑥² + 9𝑥 + 4
Example Problem (STAAR Algebra 1)
Which expression is equivalent to (ℎ² + 9ℎ − 1)(−4ℎ + 3)?
Solution:
(ℎ² + 9ℎ − 1)(−4ℎ + 3)
= ℎ²(−4ℎ) + 3ℎ² + 9ℎ(−4ℎ) + 27ℎ + 4ℎ − 3
= −4ℎ³ + 3ℎ² − 36ℎ² + 27ℎ + 4ℎ − 3
= −4ℎ³ − 33ℎ² + 31ℎ − 3
Summary
- Multiply coefficients with coefficients, variables with variables
- Remove parentheses using the distributive property
- Simplify exponents using exponent rules
- Always combine like terms at the end
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