Question

essay 20 points

1. $-3x^{2}+7-x^{3}-2x+9x^{4}$ ---------> $-8x -15x^{3}+3 -x^{2}-11x^{4}$?

Explanation:

Step1: Define the unknown polynomial

Let the unknown polynomial be $P(x)$. The equation is:
$$-3x^2 + 7 - x^3 - 2x + 9x^4 + P(x) = -8x - 15x^3 + 3 - x^2 - 11x^4$$

Step2: Rearrange to solve for $P(x)$

Isolate $P(x)$ by moving the left polynomial to the right:
$$P(x) = (-8x - 15x^3 + 3 - x^2 - 11x^4) - (-3x^2 + 7 - x^3 - 2x + 9x^4)$$

Step3: Distribute the negative sign

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

Step4: Combine like terms

Group by powers of $x$:

• $x^4$ terms: $-11x^4 -9x^4 = -20x^4$
• $x^3$ terms: $-15x^3 +x^3 = -14x^3$
• $x^2$ terms: $-x^2 +3x^2 = 2x^2$
• $x$ terms: $-8x +2x = -6x$
• constant terms: $3 -7 = -4$

Combine all:
$$P(x) = -20x^4 -14x^3 +2x^2 -6x -4$$

Answer:

$\boldsymbol{-20x^4 -14x^3 +2x^2 -6x -4}$