The polynomial 4x^2 - 8x - 5 is subtracted from the polynomial 7x^2 - 3x + 4. The simplifi… — ACT

The polynomial $4x^2 - 8x - 5$ is subtracted from the polynomial $7x^2 - 3x + 4$ . The simplified difference can be expressed as $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constant integers. What is the value of the coefficient $b$ ?

Answer: 5

Answer

The coefficient of the

x

term,

b

, is 5.

Subtracting

4x^2 - 8x - 5

from

7x^2 - 3x + 4

yields

(7x^2 - 4x^2) + (-3x - (-8x)) + (4 - (-5)) = 3x^2 + 5x + 9

. The coefficient of the linear term

x

5

Step-by-Step Solution

Set up the subtraction expression.

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

Subtracting the second polynomial from the first requires enclosing the second polynomial in parentheses to apply the subtraction to all terms.

Distribute the negative sign to all terms inside the parentheses.

7x^2 - 3x + 4 - 4x^2 + 8x + 5

Distributing the subtraction sign flips the sign of each term: positive terms become negative, and negative terms become positive.

Group and combine like terms.

3x^2 + 5x + 9

Combine the coefficients of matching variable parts:

(7 - 4)x^2 = 3x^2

(-3 + 8)x = 5x

, and

4 - (-5) = 9

Identify the coefficient

b

corresponding to the

x

term.

b = 5

Comparing the simplified expression

3x^2 + 5x + 9

ax^2 + bx + c

shows that the coefficient of

x

is 5.

Key Concept

Polynomial Subtraction and Combining Like Terms

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