Question

answer the following questions. (1) choose the best answer. a shortcut method of dividing a polynomial by a linear polynomial by using only the coefficients of the terms of the polynomial is called __. shortcut division polynomial division coefficient division synthetic division (2) choose the correct equation represented by the division problem. (x - 2) ÷ (3x³ + 5x² - 14x - 18) = 3x² + 11x + 8 - 2 (3x³ + 5x² - 14x - 18) ÷ (x - 2) = 3x³ + 11x + 8 - \frac{2}{x - 2} (3x³ + 5x² - 14x - 18) ÷ (x + 2) = 3x³ + 11x² + 8x - 2 (3x³ + 5x² - 14x - 18) ÷ (x - 2) = 3x² + 11x + 8 - 2

Explanation:

1. For question 1: The described method that uses only coefficients to divide a polynomial by a linear polynomial is defined as synthetic division.
2. For question 2: In synthetic division, the number on the left is the root of the divisor $(x - k)$, so here $k=2$, meaning divisor is $(x-2)$. The top row are coefficients of the dividend $3x^3 +5x^2 -14x -18$. The bottom row (excluding last number) are quotient coefficients $3x^2+11x+8$, last number is remainder $-2$. So the division is $(3x^3 + 5x^2 -14x -18) \div (x - 2) = 3x^2 + 11x + 8 - \frac{2}{x-2}$.

Answer:

1. synthetic division
2. $(3x^3 + 5x^2 -14x -18) \div (x - 2) = 3x^2 + 11x + 8 - \frac{2}{x-2}$