Fix the mistake: Calculations with Polynomials

Polynomial Calculations ① Multiplication and Division of Monomials

Before beginning polynomial calculations, first master multiplication and division of monomials. We will also gradually use the laws of exponents.

Multiplication of Monomials

Multiply the coefficients and the variables separately. Combine the same variables into powers.
　Example: 3x × 4y ＝ 12xy、2a × 5a ＝ 10a²、(−3x)×(4x²) ＝ −12x³

Example ①
Calculate the following.
(1) 4x × (−3y)　(2) 2a² × 5a　(3) (−6xy) × (−2x)
Answer:
(1) −12xy、(2) 10a³、(3) 12x²y

Division of Monomials

Write the expression as a fraction and cancel the same variables.
　Example: 6x ÷ 2 ＝ 3x、12a²b ÷ 4ab ＝ 3a、(−15xy) ÷ 5x ＝ −3y

Example ②
Calculate the following.
(1) 8x²y ÷ 2xy　(2) (−12a²b) ÷ (−4a)
Answer:(1) 4x、(2) 3ab

Combined Multiplication and Division

When multiplication and division are mixed in an expression, change division into a fraction, or multiplication by the reciprocal, before calculating.

Example ③
Calculate the following: 6x²y × 4y ÷ 8xy
Answer:(6x²y × 4y) / (8xy) ＝ 24x²y² / 8xy ＝ 3xy。
Key Points
  • Monomial × monomial: multiply the coefficients and variables separately
  • Monomial ÷ monomial: write it as a fraction and cancel
  • Mixed multiplication and division can be calculated at once as a fraction
  • Check all signs together at the end

Practice Problems

  1. (−5a) × 3a²
  2. 18x²y ÷ (−6x)
  3. 4ab × 6a ÷ 3b

Answers and Explanations

  1. Answer:−15a³
    Explanation: (−5)×3 = −15、a×a² = a³。
  2. Answer:−3xy
    Explanation: 18x²y / (−6x) = −3xy。
  3. Answer:8a²
    Explanation: 24a²b / 3b = 8a²。

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