We will learn how to multiply a polynomial by a monomial. Use the distributive property.
Polynomial × Monomial
By the distributive property, multiply the monomial by each term inside the parentheses. a(b + c) ＝ ab + ac
Example ①
(1) 3x(2x − 5) (2) (−2a)(3a + 4b − 1)
Answer:(1) 6x² − 15x、(2) −6a² − 8ab + 2a
(1) 3x(2x − 5) (2) (−2a)(3a + 4b − 1)
Answer:(1) 6x² − 15x、(2) −6a² − 8ab + 2a
Polynomial ÷ Monomial
Divide each term by the monomial. (a + b) ÷ c ＝ a/c + b/c
Example ②
(8x²y − 6xy²) ÷ 2xy
Answer:8x²y/(2xy) − 6xy²/(2xy) ＝ 4x − 3y。
(8x²y − 6xy²) ÷ 2xy
Answer:8x²y/(2xy) − 6xy²/(2xy) ＝ 4x − 3y。
Calculations Involving Parentheses
Example ③
2x(3x − 1) − 3x(x − 4)
Answer:Distribute → 6x² − 2x − 3x² + 12x. Combine like terms → 3x² + 10x。
2x(3x − 1) − 3x(x − 4)
Answer:Distribute → 6x² − 2x − 3x² + 12x. Combine like terms → 3x² + 10x。
Key Points
- Polynomial × monomial: multiply each term using the distributive property
- Polynomial ÷ monomial: divide each term by the monomial
- Combine variables multiplied together into powers
- Remove parentheses → combine like terms
Practice Problems
- 4a(2a − 3b)
- (10x²y + 15xy²) ÷ 5xy
- 3a(a + 2) − 2a(a − 3)
Answers and Explanations
- Answer:8a² − 12ab
Explanation: 4a×2a=8a²、4a×(−3b)=−12ab。 - Answer:2x + 3y
Explanation: 10x²y/5xy=2x、15xy²/5xy=3y。 - Answer:a² + 12a
Explanation: 3a²+6a − 2a²+6a = a²+12a。
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Polynomial Calculations ① Multiplication and Division of Monomials
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Polynomial Calculations ③ Evaluating Expressions and Efficient Substitution
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