We will learn how to multiply polynomials by numbers and divide them by numbers. The basic principle is the distributive property. This is the Grade 7 calculation of linear expressions, with only more terms.
Multiplying Polynomials by Numbers
When multiplying a polynomial by a number, multiply every term inside the parentheses (the distributive property).
a × (b + c + d) ＝ ab + ac + ad
Example ①
Calculate the following.
(1) 3(2x² − 4x + 5) (2) (−2)(3a − 5b + 1)
Answer:
(1) 3×2x² + 3×(−4x) + 3×5 ＝ 6x² − 12x + 15
(2) (−2)×3a + (−2)×(−5b) + (−2)×1 ＝ −6a + 10b − 2
Calculate the following.
(1) 3(2x² − 4x + 5) (2) (−2)(3a − 5b + 1)
Answer:
(1) 3×2x² + 3×(−4x) + 3×5 ＝ 6x² − 12x + 15
(2) (−2)×3a + (−2)×(−5b) + (−2)×1 ＝ −6a + 10b − 2
Dividing Polynomials by Numbers
When dividing a polynomial by a number, calculate by dividing each term by the number or by multiplying by its reciprocal.
(A + B) ÷ c ＝ A/c + B/c ＝ (1/c)(A + B)
Example ②
Calculate the following: (12x² − 8x + 4) ÷ (−4)
Answer: Divide each term by −4: 12x² ÷ (−4) ＝ −3x², −8x ÷ (−4) ＝ +2x, 4 ÷ (−4) ＝ −1.
Therefore, −3x² + 2x − 1.
Calculate the following: (12x² − 8x + 4) ÷ (−4)
Answer: Divide each term by −4: 12x² ÷ (−4) ＝ −3x², −8x ÷ (−4) ＝ +2x, 4 ÷ (−4) ＝ −1.
Therefore, −3x² + 2x − 1.
Key Point
- Polynomial×number: use the distributive property to multiply every term inside the parentheses by the number
- Polynomial÷number: divide each term by the number or multiply by its reciprocal
- When multiplying or dividing by a negative number, the signs of all terms change
- A fractional expression can be calculated by distributing 1/(denominator) to each term
Practice Problems
- Calculate the following: (−4)(2x² + 3x − 5)
- Calculate the following: (15a − 10b + 5) ÷ 5
- Calculate the following: (6x − 9)/3
Answer and Explanation
- Answer:−8x² − 12x + 20
Explanation: (−4)×2x²=−8x², (−4)×3x=−12x, (−4)×(−5)=+20. - Answer:3a − 2b + 1
Explanation: 15a÷5=3a, −10b÷5=−2b, 5÷5=1. - Answer:2x − 3
Explanation: Divide each term by 3. 6x÷3=2x, −9÷3=−3.