Using long division, determine which of the following polynomials divides evenly for a divisor of x+4

Question

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Answer 1

To determine which of the polynomials divides evenly for a divisor of x + 4, we can use long division.

Let's start with the first polynomial:

x^3 + 3x^2 - 4x - 12

To divide this polynomial by x + 4, we first divide the highest degree term:

x^3 ÷ (x + 4) = x^2

Now, we multiply x + 4 by x^2:

x^2 * (x + 4) = x^3 + 4x^2

Next, we subtract this from the original polynomial:

(x^3 + 3x^2 - 4x - 12) - (x^3 + 4x^2) = -x^2 - 4x - 12

Now, we bring down the next term, which is -x^2:

- x^2 - 4x - 12

To continue the division, we divide -x^2 by x + 4:

- x^2 ÷ (x + 4) = -1x

Now, we multiply x + 4 by -1x:

-1x * (x + 4) = -x^2 - 4x

Next, we subtract this from the previous result:

(-x^2 - 4x - 12) - (-x^2 - 4x) = -12

As there are no more terms to bring down, the division is finished. The quotient is x^2 - 1x and the remainder is -12.

Therefore, the polynomial x^3 + 3x^2 - 4x - 12 does not divide evenly for a divisor of x + 4.