Questions: Question 14 Use long division to determine whether or not the binomial is a factor of f(x). f(x)=x^3-3x^2-40x+84; x+6

Solution Steps

We begin by dividing the polynomial \( f(x) = x^{3} - 3x^{2} - 40x + 84 \) by the binomial \( x + 6 \).

1. Divide \( x^{3} \) by \( x \) to get \( x^{2} \).
2. Multiply \( x^{2} \) by \( x + 6 \) to get \( x^{3} + 6x^{2} \).
3. Subtract this from the original polynomial, resulting in \( -9x^{2} - 40x + 84 \).

Next, we continue the division:

1. Divide \( -9x^{2} \) by \( x \) to get \( -9x \).
2. Multiply \( -9x \) by \( x + 6 \) to get \( -9x^{2} - 54x \).
3. Subtract this from the current polynomial, resulting in \( 14x + 84 \).

Finally, we complete the division:

1. Divide \( 14x \) by \( x \) to get \( 14 \).
2. Multiply \( 14 \) by \( x + 6 \) to get \( 14x + 84 \).
3. Subtract this from the current polynomial, resulting in a remainder of \( 0 \).

The quotient from the division is: \[ x^{2} - 9x + 14 \] The remainder is: \[ 0 \]

Since the remainder is \( 0 \), this indicates that \( x + 6 \) is a factor of \( f(x) \).

Final Answer