Question

explain what the colored numbers represent in the synthetic division shown.
-3 | 1 -2 -9 18
| -3 15 -18
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1 -5 6 0

the colored numbers represent the { } of the synthetic division problem. the blue number to the right is the { }, and each of the red numbers is the { } of each term.

options: degree, coefficient, quotient, remainder

Explanation:

Step1: Recall Synthetic Division Basics

In synthetic division, we divide a polynomial by a linear factor \(x - c\) (here \(c=-3\) since we have \(-3\) outside the division box). The numbers at the bottom (colored) relate to the quotient and remainder. The last number (blue, 0 here) is the remainder. The other colored numbers (1, -5, 6) are coefficients of the quotient polynomial. The quotient is the result of the division (excluding the remainder), and each red number is a coefficient of the quotient terms. The first blank: "quotient" (since they form the quotient polynomial's coefficients). The second blank (blue number): "remainder" (as the last number in synthetic division is the remainder). The third blank (red numbers): "coefficient" (each red number is a coefficient of the quotient's terms).

Answer:

The colored numbers represent the \(\boldsymbol{\text{quotient}}\) of the synthetic division problem. The blue number to the right is the \(\boldsymbol{\text{remainder}}\), and each of the red numbers is the \(\boldsymbol{\text{coefficient}}\) of each term.