Expanding an expression means removing parentheses and changing it into a sum. Let us first master the basic distributive law.
Distributive Law
a(b + c) = ab + ac. To multiply a monomial by a linear expression, multiply it by each term inside the parentheses.
- 3x(2x − 5) = 6x² − 15x
- −2a(a² + 3a − 4) = −2a³ − 6a² + 8a
Products of Polynomials
(a + b)(c + d) = ac + ad + bc + bd. Add the products of every pair of terms.
Example ①
Expand (2x + 3)(x − 4).
Answer: 2x·x + 2x·(−4) + 3·x + 3·(−4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12
Expand (2x + 3)(x − 4).
Answer: 2x·x + 2x·(−4) + 3·x + 3·(−4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12
Products Containing Three or More Terms
Even when there are more terms, add the products of all combinations of terms.
Example ②
Expand (x + 2)(x² − 3x + 1).
Answer: x(x²−3x+1) + 2(x²−3x+1) = x³−3x²+x + 2x²−6x+2 = x³ − x² − 5x + 2
Expand (x + 2)(x² − 3x + 1).
Answer: x(x²−3x+1) + 2(x²−3x+1) = x³−3x²+x + 2x²−6x+2 = x³ − x² − 5x + 2
Key Point
- Check coefficients and signs one by one when using the distributive law
- For products of polynomials, “add the products of every pair of terms”
- Combine like terms at the end
Practice Problems
- Expand −3x(2x² − 4x + 1).
- Expand (x + 5)(2x − 3).
- Expand (a − 2)(a² + 3a − 5).
Answers and Explanations
- Answer: −6x³ + 12x² − 3x
Explanation: Multiply −3x by each term. - Answer: 2x² + 7x − 15
Explanation: 2x² − 3x + 10x − 15 = 2x² + 7x − 15. - Answer: a³ + a² − 11a + 10
Explanation: a³+3a²−5a − 2a²−6a+10 = a³+a²−11a+10.