Polynomials
Algebra and functions
- AS
- Non-calculator
- 8 questions
- 37 marks
These are original practice questions for the Edexcel A-level Pure content, not a past paper. This site is independent and not affiliated with Pearson.
-
Divide $x^3 - 6x^2 + 11x - 6$ by $x - 1$, and factorise the quotient.
3 marks
-
Show that $x - 2$ is a factor of $x^3 - 3x^2 - 4x + 12$.
3 marks
-
Factorise $x^3 - 3x^2 - 4x + 12$ completely.
4 marks
-
Find the remainder when $x^3 + 2x^2 - 5x + 1$ is divided by $x + 3$.
4 marks
-
The remainder when $x^3 + ax + b$ is divided by $x - 1$ is $4$. The remainder when it is divided by $x + 2$ is $-2$. Find $a$ and $b$.
5 marks
-
The polynomial $x^3 + px^2 + qx - 6$ has factors $x - 1$ and $x + 2$. Find $p$ and $q$, and factorise the polynomial completely.
5 marks
-
Factorise $2x^3 + x^2 - 5x + 2$ completely.
6 marks
-
The polynomial $f(x) = x^3 + ax^2 + bx + c$ has factors $x - 1$, $x - 2$ and $x + 3$. Find $a$, $b$ and $c$, and find the remainder when $f(x)$ is divided by $x - 4$.
7 marks
Mark scheme
-
Answer. The quotient is $x^2 - 5x + 6 = (x - 2)(x - 3)$.
- By division, or by the factor theorem and comparing coefficients, the quotient is $x^2 - 5x + 6$.
- $x^2 - 5x + 6 = (x - 2)(x - 3)$.
-
Answer. $(x - 2)$ is a factor because $f(2) = 0$.
- $f(2) = 8 - 12 - 8 + 12 = 0$.
- By the factor theorem, $x - 2$ is a factor.
-
Answer. $(x - 2)(x - 3)(x + 2)$
- Since $f(2) = 0$, divide by $x - 2$ to get $x^2 - x - 6$.
- $x^2 - x - 6 = (x - 3)(x + 2)$.
-
Answer. $7$
- The remainder is $f(-3)$.
- $f(-3) = -27 + 18 + 15 + 1 = 7$.
-
Answer. $a = -1$ and $b = 4$
- $f(1) = 1 + a + b = 4$, so $a + b = 3$.
- $f(-2) = -8 - 2a + b = -2$, so $-2a + b = 6$.
- Subtract to get $3a = -3$, so $a = -1$ and $b = 4$.
-
Answer. $p = 4$, $q = 1$, and the polynomial is $(x - 1)(x + 2)(x + 3)$.
- $f(1) = 0$ gives $p + q = 5$.
- $f(-2) = 0$ gives $2p - q = 7$.
- $p = 4$ and $q = 1$.
- The third factor is $x + 3$.
-
Answer. $(x - 1)(2x - 1)(x + 2)$
- $f(1) = 0$, so $x - 1$ is a factor.
- Division gives $2x^2 + 3x - 2$.
- $2x^2 + 3x - 2 = (2x - 1)(x + 2)$.
-
Answer. $a = 0$, $b = -7$, $c = 6$, and the remainder is $42$.
- $f(x) = (x - 1)(x - 2)(x + 3) = x^3 - 7x + 6$.
- So $a = 0$, $b = -7$ and $c = 6$.
- The remainder is $f(4) = 64 - 28 + 6 = 42$.