Question

(a) complete this synthetic division table.
-4) 3 10 -9 0 12
□ □ □ □
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(b) write your answer in the following form: quotient+$\frac{remainder}{x+4}$,
$\frac{3x^{4}+10x^{3}-9x^{2}+12}{x+4}=□+\frac{□}{x+4}$

Explanation:

Step1: Bring down leading coefficient

Bring down 3 to the bottom row.

Step2: Multiply by divisor, add to next term

$3 \times (-4) = -12$; $10 + (-12) = -2$

Step3: Multiply by divisor, add to next term

$-2 \times (-4) = 8$; $-9 + 8 = -1$

Step4: Multiply by divisor, add to next term

$-1 \times (-4) = 4$; $0 + 4 = 4$

Step5: Multiply by divisor, add to next term

$4 \times (-4) = -16$; $12 + (-16) = -4$

Answer:

(a) Completed synthetic division table:

-4)  3   10   -9    0    12
        -12    8   -4   -16
      ----------------------
      3   -2   -1    4    -4

(b)

$\frac{3x^4 + 10x^3 - 9x^2 + 12}{x+4} = 3x^3 - 2x^2 - x + 4 + \frac{-4}{x+4}$