Question

5 essay 20 points
7)
$5x^3 - 3x - 15x^4 - x^2 - 8 \
ightarrow -3 + 3x^2 + x^4 + 13x + 7x^3?$

Explanation:

Step1: Rearrange terms by degree

$$-15x^4 + 5x^3 - x^2 - 3x - 8$$

Step2: Set up the addition equation

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

Step3: Combine like terms for $x^4$

$$-15x^4 + x^4 = -14x^4$$

Step4: Combine like terms for $x^3$

$$5x^3 + 7x^3 = 12x^3$$

Step5: Combine like terms for $x^2$

$$-x^2 + 3x^2 = 2x^2$$

Step6: Combine like terms for $x$

$$-3x + 13x = 10x$$

Step7: Combine constant terms

$$-8 + (-3) = -11$$

Step8: Assemble the final polynomial

$$-14x^4 + 12x^3 + 2x^2 + 10x - 11$$

Answer:

$\boldsymbol{-14x^4 + 12x^3 + 2x^2 + 10x - 11}$