Question

Find the quotient and remainder when

−4x^3 + 2x^2 − 4 is divided by x−1.
a. Quotient: −4x^2 −2x−2, Remainder: −2
b. Quotient: −4x^2 +2x+2, Remainder: −2
c. Quotient: −4x^2 +2x−2, Remainder: −4
d. Quotient: −4x^2 −2x+2, Remainder: −4

Answer 1

The quotient when −4x3 + 2x2 − 4 is divided by x−1 is −4x2 + 2x + 6, and the remainder is −2, which corresponds to option b.

Step-by-step explanation:

To find the quotient and remainder when −4x3 + 2x2 − 4 is divided by x−1, use long division or synthetic division. Here are the steps for polynomial division:

1. Divide the first term of the dividend (−4x3) by the first term of the divisor (x), which gives −4x2.
2. Multiply the entire divisor by this term and subtract it from the dividend.
3. Bring down the next term from the dividend and repeat the process until all terms have been brought down.

Performing these steps yields:

• Quotient from step 1: −4x2
• x(−4x2) = −4x3, and (1)(−4x2) = −4x2. This gives us −4x3 + −4x2 to subtract from the dividend.
• Subtracting, we get 2x2 − (−4x2) = 6x2, bring down the −4, repeat division.
• Divide 6x2 by x which is 6x, now multiply the divisor by this term: x(6x) = 6x2 and (1) (6x)= 6x. Differences are 0 and −4 − 6x = −6x.
• Divide −6x by x which results in −6, followed by subtracting the result of the divisor multiplied by −6.

The result is a quotient of −4x2 + 2x + 6 and a remainder of −4 − (6)(1) = −4 − 6 = −2.

Therefore, the correct answer to the question is:

• Quotient: −4x2 + 2x + 6,
• Remainder: −2.

This corresponds to option b: Quotient: −4x2 + 2x + 6, Remainder: −2.