1. **Problem statement:** Use synthetic division to divide the polynomial $f(x) = 4x^3 - x^2 + 4$ by $x - 2$ and find the quotient and remainder. 2. **Formula and rules:** Synthetic division is a shortcut method for dividing a polynomial by a linear divisor of the form $x - c$. We use the value $c = 2$ here. 3. **Set up synthetic division:** Write the coefficients of $f(x)$ in descending order of powers: $4$ (for $x^3$), $-1$ (for $x^2$), $0$ (for $x$ term, missing), and $4$ (constant term). 4. **Perform synthetic division:** - Bring down the first coefficient: $4$. - Multiply by $2$: $4 \times 2 = 8$. - Add to next coefficient: $-1 + 8 = 7$. - Multiply by $2$: $7 \times 2 = 14$. - Add to next coefficient: $0 + 14 = 14$. - Multiply by $2$: $14 \times 2 = 28$. - Add to last coefficient: $4 + 28 = 32$. 5. **Interpret results:** The numbers $4, 7, 14$ are the coefficients of the quotient polynomial of degree 2, and $32$ is the remainder. 6. **Write quotient and remainder:** $$\text{Quotient} = 4x^2 + 7x + 14$$ $$\text{Remainder} = 32$$ **Final answer:** When $f(x) = 4x^3 - x^2 + 4$ is divided by $x - 2$, the quotient is $4x^2 + 7x + 14$ and the remainder is $32$.