Below are practice questions on synthetic division of polynomials. Each problem asks you to determine the quotient \(Q\) and the remainder \(R\). Answers are provided at the end of the page.
For additional practice, see this reference on polynomials.
Use synthetic division to find the quotient \(Q\) and remainder \(R\) when dividing
\[ 2x^2 + x \quad \text{by} \quad x - 1 \]Use synthetic division to find the quotient \(Q\) and remainder \(R\) when dividing
\[ 4x^3 + x^2 - 2x + 3 \quad \text{by} \quad x + 2 \]Use synthetic division to find the quotient \(Q\) and remainder \(R\) when dividing
\[ 7x^5 - 2x^4 - 5x^3 + x^2 - 3x + 5 \quad \text{by} \quad x - 2 \]Use synthetic division to find the quotient \(Q\) and remainder \(R\) when dividing
\[ \frac{1}{2}x^3 - \frac{1}{3}x^2 - \frac{3}{2}x + \frac{1}{3} \quad \text{by} \quad x - \frac{1}{2} \]Use synthetic division to find the quotient \(Q\) and remainder \(R\) when dividing
\[ 0.3x^3 - 0.2x^2 - 0.1x + 0.3 \quad \text{by} \quad x - 0.2 \]Use synthetic division and the Remainder Theorem to determine \(f(-2)\), given
\[ f(x) = 5x^4 - 4x^3 - x^2 + 5 \]Use synthetic division and the Remainder Theorem to determine \(f\!\left(\frac{1}{2}\right)\), given
\[ f(x) = -3x^5 + x^3 - 2x^2 + x - 3 \]Use synthetic division and the Remainder Theorem to determine \(f(0.2)\), given
\[ f(x) = 0.1x^3 - 0.6x^2 - 0.6 \]Use synthetic division to find the quotient \(Q\) and remainder \(R\) when dividing
\[ f(x) = x^2 + 2x + 3 \quad \text{by} \quad x - i \]where \( i = \sqrt{-1} \) is the imaginary unit.
Use synthetic division to find the quotient \(Q\) and remainder \(R\) when dividing
\[ f(x) = x^5 + x^4 + x^3 + 2x - 5 \quad \text{by} \quad x + i \]where \( i = \sqrt{-1} \).
1. \( Q = 2x + 3,\; R = 3 \)
2. \( Q = 4x^2 - 7x + 12,\; R = -21 \)
3. \( Q = 7x^4 + 12x^3 + 19x^2 + 39x + 75,\; R = 155 \)
4. \[ Q = \frac{1}{2}x^2 - \frac{1}{2}x - \frac{37}{24}, \quad R = -\frac{7}{16} \]
5. \[ Q = 0.3x^2 - 0.14x - 0.128, \quad R = 0.2744 \]
6. \( f(-2) = 113 \)
7. \( f\!\left(\frac{1}{2}\right) = \frac{95}{32} \)
8. \( f(0.2) = -0.6232 \)
9. \[ Q = x + 2 + i, \quad R = 2 + 2i \]
10. \[ Q = x^4 + (1 - i)x^3 - ix^2 - x + 2 + i, \quad R = -2i - 4 \]