Express \( f(x) \) in the form \( f(x)=(x-k) q(x)+r \) for the given value of \( k \). \( f(x)=5 x^{3}+x^{2}+x-4, k=-1 \)

To express \( f(x) = 5x^3 + x^2 + x - 4 \) in the form \( f(x) = (x - k)q(x) + r \) for \( k = -1 \), we will perform polynomial long division of \( f(x) \) by \( x - (-1) = x + 1 \). 1. **Set up the division**: We want to divide \( f(x) \) by \( x + 1 \). 2. **Perform the division**: - Divide the leading term of \( f(x) \) (which is \( 5x^3 \)) by the leading term of \( x + 1 \) (which is \( x \)): \[ \frac{5x^3}{x} = 5x^2 \] - Multiply \( 5x^2 \) by \( x + 1 \): \[ 5x^2(x + 1) = 5x^3 + 5x^2 \] - Subtract this from \( f(x) \): \[ (5x^3 + x^2 + x - 4) - (5x^3 + 5x^2) = (x^2 - 5x^2) + x - 4 = -4x^2 + x - 4 \] 3. **Repeat the process**: - Divide the leading term \( -4x^2 \) by \( x \): \[ \frac{-4x^2}{x} = -4x \] - Multiply \( -4x \) by \( x + 1 \): \[ -4x(x + 1) = -4x^2 - 4x \] - Subtract this from the current remainder: \[ (-4x^2 + x - 4) - (-4x^2 - 4x) = (x + 4x) - 4 = 5x - 4 \] 4. **Repeat again**: - Divide the leading term \( 5x \) by \( x \): \[ \frac{5x}{x} = 5 \] - Multiply \( 5 \) by \( x + 1 \): \[ 5(x + 1) = 5x + 5 \] - Subtract this from the current remainder: \[ (5x - 4) - (5x + 5) = -4 - 5 = -9 \] 5. **Final result**: - The division gives us: \[ f(x) = (x + 1)(5x^2 - 4x + 5) - 9 \] Thus, we can express \( f(x) \) in the desired form: \[ f(x) = (x + 1)(5x^2 - 4x + 5) - 9 \] In conclusion, the expression for \( f(x) \) is: \[ f(x) = (x + 1)q(x) + r \quad \text{where } q(x) = 5x^2 - 4x + 5 \text{ and } r = -9. \]