Question

use what you know about addition and subtraction of polynomials to find the missing term represented by the question mark in each equation.

\\(-4x^2 + 6x + 3) + (8x + 5 + ?) = 2x^2 + 14x + 8\\

\\(? = \\)

\\(-b^3 + 3b^2 + 8) - (? - 5b^2 - 9) = 5b^3 + 8b^2 + 17\\

\\(? = \\)

Explanation:

⚡ Using what you learned: adding and subtracting polynomials

Step 1: Solve the first equation

Let the first missing term be \( Y \).

$$ (-4x^2 + 6x + 3) + (8x + 5 + Y) = 2x^2 + 14x + 8 $$

Combine the known terms on the left side:

$$ -4x^2 + (6x + 8x) + (3 + 5) + Y = 2x^2 + 14x + 8 $$

$$ -4x^2 + 14x + 8 + Y = 2x^2 + 14x + 8 $$

Subtract \( (-4x^2 + 14x + 8) \) from both sides to isolate \( Y \):

$$ Y = (2x^2 + 14x + 8) - (-4x^2 + 14x + 8) $$

$$ Y = 2x^2 + 14x + 8 + 4x^2 - 14x - 8 $$

$$ Y = 6x^2 $$

Step 2: Solve the second equation

Let the second missing term be \( Z \).

$$ (-b^3 + 3b^2 + 8) - (Z - 5b^2 - 9) = 5b^3 + 8b^2 + 17 $$

Distribute the negative sign:

$$ -b^3 + 3b^2 + 8 - Z + 5b^2 + 9 = 5b^3 + 8b^2 + 17 $$

Combine the known terms on the left side:

$$ -b^3 + (3b^2 + 5b^2) + (8 + 9) - Z = 5b^3 + 8b^2 + 17 $$

$$ -b^3 + 8b^2 + 17 - Z = 5b^3 + 8b^2 + 17 $$

Subtract \( (-b^3 + 8b^2 + 17) \) from both sides to isolate \( -Z \):

$$ -Z = (5b^3 + 8b^2 + 17) - (-b^3 + 8b^2 + 17) $$

$$ -Z = 5b^3 + 8b^2 + 17 + b^3 - 8b^2 - 17 $$

$$ -Z = 6b^3 $$

Multiply both sides by \( -1 \):

$$ Z = -6b^3 $$

Answer:

For the first equation:

$$ ? = 6x^2 $$

For the second equation:

$$ ? = -6b^3 $$