The first topic in ninth grade is “expanding expressions.” Use the distributive property learned in eighth grade to remove parentheses (expansion).
Monomial×Polynomial
Use the distributive property a(b+c) = ab + ac directly.
Example: 2x(3x − 4) = 2x×3x + 2x×(−4) = 6x² − 8x
Polynomial×Polynomial
To expand (a + b)(c + d), multiply each term in the first parentheses by each term in the second parentheses once, and add all the results.
(a + b)(c + d) = ac + ad + bc + bd
- (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6
- (2x − 1)(x + 4) = 2x² + 8x − x − 4 = 2x² + 7x − 4
Example ①
Expand the following.
(1) −3a(2a − 5) (2) (x − 4)(x + 6)
Answer:
(1) −3a×2a + (−3a)×(−5) = −6a² + 15a
(2) x×x + x×6 + (−4)×x + (−4)×6 = x² + 6x − 4x − 24 = x² + 2x − 24
Expand the following.
(1) −3a(2a − 5) (2) (x − 4)(x + 6)
Answer:
(1) −3a×2a + (−3a)×(−5) = −6a² + 15a
(2) x×x + x×6 + (−4)×x + (−4)×6 = x² + 6x − 4x − 24 = x² + 2x − 24
Combining Like Terms
After expanding, always combine like terms. Add the terms containing x together and the constant terms together.
Example ②
Expand (2x + 3)(3x − 5).
Answer: 2x×3x + 2x×(−5) + 3×3x + 3×(−5) = 6x² − 10x + 9x − 15 = 6x² − x − 15
Expand (2x + 3)(3x − 5).
Answer: 2x×3x + 2x×(−5) + 3×3x + 3×(−5) = 6x² − 10x + 9x − 15 = 6x² − x − 15
Key Points
- Monomial×polynomial uses the distributive property: a(b+c) = ab+ac
- Polynomial×polynomial: add all products of “each term in the first × each term in the second” (4 terms).
- Always combine like terms after expanding.
- Be careful with signs. Perform multiplication involving negative numbers accurately.
Practice Problems
- Expand 4x(2x − 3y).
- Expand (x − 5)(x + 2).
- Expand (3a − 2)(2a + 5).
Answer and Explanation
- Answer: 8x² − 12xy
Explanation: 4x×2x = 8x², 4x×(−3y) = −12xy. - Answer: x² − 3x − 10
Explanation: x² + 2x − 5x − 10 = x² − 3x − 10. - Answer: 6a² + 11a − 10
Explanation: 6a² + 15a − 4a − 10 = 6a² + 11a − 10.