Questions: Use the remainder theorem and synthetic division to find f(k) for the given f(x)=-2x^3-13x^2-21x+5 ; k=-4 f(-4)=

Solution Steps

We start by dividing the polynomial \( f(x) = -2x^{3} - 13x^{2} - 21x + 5 \) by the divisor \( x + 4 \).

1. Divide the leading term \( -2x^{3} \) by \( x \), resulting in \( -2x^{2} \).
2. Multiply \( -2x^{2} \) by \( x + 4 \) to get \( -2x^{3} - 8x^{2} \).
3. Subtract this from the original polynomial, yielding the new polynomial \( -5x^{2} - 21x + 5 \).

Next, we continue the division process with the new polynomial \( -5x^{2} - 21x + 5 \).

1. Divide the leading term \( -5x^{2} \) by \( x \), resulting in \( -5x \).
2. Multiply \( -5x \) by \( x + 4 \) to get \( -5x^{2} - 20x \).
3. Subtract this from the current polynomial, yielding \( -x + 5 \).

We perform one last division step with the polynomial \( -x + 5 \).

1. Divide the leading term \( -x \) by \( x \), resulting in \( -1 \).
2. Multiply \( -1 \) by \( x + 4 \) to get \( -x - 4 \).
3. Subtract this from the current polynomial, yielding a remainder of \( 9 \).

The complete division can be expressed as:

\[ \frac{-2x^{3} - 13x^{2} - 21x + 5}{x + 4} = -2x^{2} - 5x - 1 + \frac{9}{x + 4} \]

According to the Remainder Theorem, the value of \( f(-4) \) is equal to the remainder obtained from the division. Thus, we find:

\[ f(-4) = 9 \]

Final Answer