Question

1. \\((k^4 + 8k^3 + 20k^2 + 5k - 30) \div (k + 3)\\)

using synthetic division with root -3 for the polynomial coefficients 1, 8, 20, 5, -30:
adding the results of synthetic division steps: the bottom row is 1, 5, 5, -10, 0. the quotient is \\(k^3 + 5k^2 + 5k - 10\\)

Explanation:

Step1: Set up synthetic division

Use root $k=-3$, coefficients $1,8,20,5,-30$

Step2: Bring down leading coefficient

$\text{New coefficient: } 1$

Step3: Multiply by root, add next coeff

$1\times(-3) + 8 = 5$

Step4: Repeat multiply-add step

$5\times(-3) + 20 = 5$

Step5: Repeat multiply-add step

$5\times(-3) + 5 = -10$

Step6: Repeat multiply-add step

$-10\times(-3) + (-30) = 0$

Step7: Form quotient polynomial

Use final coefficients for degree $3$ polynomial

Answer:

$k^3 + 5k^2 + 5k - 10$