1. **State the problem:** We are asked to divide the polynomial function $f(x) = 4x^4 - 33x^2 - 9x + 20$ by $p(x) = x - 3$. 2. **Recall the division formula:** When dividing a polynomial $f(x)$ by a linear divisor $p(x) = x - a$, the division can be expressed as: $$f(x) = (x - a)q(x) + r$$ where $q(x)$ is the quotient polynomial and $r$ is the remainder (a constant). 3. **Use synthetic division:** Since $p(x) = x - 3$, we use $a = 3$ for synthetic division. 4. **Set up coefficients:** The polynomial $f(x)$ is $4x^4 + 0x^3 - 33x^2 - 9x + 20$ (note the missing $x^3$ term has coefficient 0). Coefficients: $[4, 0, -33, -9, 20]$ 5. **Perform synthetic division with $a=3$:** - Bring down 4. - Multiply 4 by 3 = 12; add to 0 = 12. - Multiply 12 by 3 = 36; add to -33 = 3. - Multiply 3 by 3 = 9; add to -9 = 0. - Multiply 0 by 3 = 0; add to 20 = 20. 6. **Interpret results:** - Quotient coefficients: $[4, 12, 3, 0]$ corresponding to $4x^3 + 12x^2 + 3x + 0$ - Remainder: $20$ 7. **Write the division result:** $$f(x) = (x - 3)(4x^3 + 12x^2 + 3x) + 20$$ 8. **Note on the graph:** The graph described is unrelated to the division problem and shows an x-intercept at $x=0$ and a roughly trapezoidal shape starting at $(0,0)$ going upwards linearly. **Final answer:** The quotient is $4x^3 + 12x^2 + 3x$ and the remainder is $20$ when dividing $f(x)$ by $x - 3$.