Question

1. what polynomial could you add to $5x^{3}-3x-15x^{4}-x^{2}-8$ to get a sum of $-3 + 3x^{5}+x^{4}+13x+7x^{2}$?
2. what polynomial could you add to $-3x^{2}+7 - x^{3}-2x + 9x^{4}$ to get a sum of $-8x-15x^{3}+3 - x^{2}-11x^{4}$?

Explanation:

Step1: Define unknown polynomial

Let the polynomial to add be $P(x)$.

Step2: Set up equation for Q7

$P(x) + 5x^3 - 3x - 15x^4 - x^2 - 8 = -3 + 3x^5 + x^4 + 13x + 7x^2$

Step3: Rearrange to solve Q7 $P(x)$

$P(x) = (-3 + 3x^5 + x^4 + 13x + 7x^2) - (5x^3 - 3x - 15x^4 - x^2 - 8)$

Step4: Simplify Q7 expression

$P(x) = 3x^5 + (x^4 + 15x^4) - 5x^3 + (7x^2 + x^2) + (13x + 3x) + (-3 + 8)$
$P(x) = 3x^5 + 16x^4 - 5x^3 + 8x^2 + 16x + 5$

Step5: Set up equation for Q8

$P(x) + (-3x^2 + 7 - x^3 - 2x + 9x^4) = -8x -15x^3 + 3 - x^2 -11x^4$

Step6: Rearrange to solve Q8 $P(x)$

$P(x) = (-8x -15x^3 + 3 - x^2 -11x^4) - (-3x^2 + 7 - x^3 - 2x + 9x^4)$

Step7: Simplify Q8 expression

$P(x) = (-11x^4 -9x^4) + (-15x^3 + x^3) + (-x^2 + 3x^2) + (-8x + 2x) + (3 - 7)$
$P(x) = -20x^4 -14x^3 + 2x^2 -6x -4$

Answer:

1. For question 7: $3x^5 + 16x^4 - 5x^3 + 8x^2 + 16x + 5$
2. For question 8: $-20x^4 -14x^3 + 2x^2 -6x -4$