Unent 5 (Chapter 4) \( \frac{c^{3}-8 c^{2}+24 c-13}{c-6} \) \( \frac{c^{3}-8 c^{2}+24 c-13}{c-6}=\square \) \( ( \) Simplify your answer. Use integers or fractions for any numbers in the expression.)

To simplify the expression \( \frac{c^{3}-8 c^{2}+24 c-13}{c-6} \), we will perform polynomial long division. 1. Divide the leading term \( c^3 \) by the leading term \( c \) to get \( c^2 \). 2. Multiply \( c^2 \) by \( c - 6 \) to get \( c^3 - 6c^2 \). 3. Subtract \( (c^3 - 6c^2) \) from \( c^3 - 8c^2 + 24c - 13 \) to get: \[ -2c^2 + 24c - 13. \] 4. Now, divide \( -2c^2 \) by \( c \) to get \( -2c \). 5. Multiply \( -2c \) by \( c - 6 \) to get \( -2c^2 + 12c \). 6. Subtract \( (-2c^2 + 12c) \) from \( -2c^2 + 24c - 13 \) to find: \[ 12c - 13. \] 7. Next, divide \( 12c \) by \( c \) to get \( 12 \). 8. Multiply \( 12 \) by \( c - 6 \) to get \( 12c - 72 \). 9. Subtract \( (12c - 72) \) from \( 12c - 13 \) to yield: \[ 59. \] Now, we have \( c^2 - 2c + 12 \) as the quotient and \( 59 \) as the remainder. Therefore, the simplified form is: \[ \frac{c^3 - 8c^2 + 24c - 13}{c - 6} = c^2 - 2c + 12 + \frac{59}{c - 6}. \]