Partial Fraction Form via Division

Questions asking to express a rational function in the form (polynomial) + (proper fraction), typically by performing polynomial division when the numerator degree equals or exceeds the denominator degree.

8 questions · Moderate -0.4

1.02k Simplify rational expressions: factorising, cancelling, algebraic division

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Edexcel C3 2008 January Q1

4 marks Moderate -0.3

Given that

$$\frac { 2 x ^ { 4 } - 3 x ^ { 2 } + x + 1 } { \left( x ^ { 2 } - 1 \right) } \equiv \left( a x ^ { 2 } + b x + c \right) + \frac { d x + e } { \left( x ^ { 2 } - 1 \right) }$$ find the values of the constants $a , b , c , d$ and $e$.
(4)

Edexcel C3 2013 June Q1

4 marks Moderate -0.5

Given that

$$\frac { 3 x ^ { 4 } - 2 x ^ { 3 } - 5 x ^ { 2 } - 4 } { x ^ { 2 } - 4 } \equiv a x ^ { 2 } + b x + c + \frac { d x + e } { x ^ { 2 } - 4 } , \quad x \neq \pm 2$$ find the values of the constants $a , b , c , d$ and $e$.
(4)

OCR C4 Q1

4 marks Moderate -0.3

$\mathrm { f } ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 13 x ^ { 2 } + 26 x - 17 } { x ^ { 2 } - 3 x + 3 }$.

Find the values of the constants $A , B , C$ and $D$ such that $$f ( x ) = x ^ { 2 } + A x + B + \frac { C x + D } { x ^ { 2 } - 3 x + 3 }$$

Edexcel C3 Q8

13 marks Standard +0.3

$f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } - 9 } { x ^ { 2 } + x - 6 }$.
1. Using algebraic division, show that
$$f ( x ) = x ^ { 2 } + A + \frac { B } { x + C }$$ where $A , B$ and $C$ are integers to be found.
By sketching two suitable graphs on the same set of axes, show that the equation $\mathrm { f } ( x ) = 0$ has exactly one real root.
Use the iterative formula $$x _ { n + 1 } = 2 + \frac { 1 } { x _ { n } ^ { 2 } + 1 } ,$$ with a suitable starting value to find the root of the equation $\mathrm { f } ( x ) = 0$ correct to 3 significant figures and justify the accuracy of your answer.

Edexcel C3 Q3

10 marks Standard +0.3

$f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 13 x ^ { 2 } + 26 x - 17 } { x ^ { 2 } - 3 x + 3 } , x \in \mathbb { R }$.
1. Find the values of the constants $A$, $B$, $C$ and $D$ such that
$$f ( x ) = x ^ { 2 } + A x + B + \frac { C x + D } { x ^ { 2 } - 3 x + 3 }$$ The point $P$ on the curve $y = \mathrm { f } ( x )$ has $x$-coordinate 1.
Show that the normal to the curve $y = \mathrm { f } ( x )$ at $P$ has the equation $$x + 5 y + 9 = 0$$

AQA C4 2005 June Q3

6 marks Moderate -0.8

Find the remainder when $2 x ^ { 3 } - x ^ { 2 } + 2 x - 2$ is divided by $2 x - 1$.
Given that $\frac { 2 x ^ { 3 } - x ^ { 2 } + 2 x - 2 } { 2 x - 1 } = x ^ { 2 } + a + \frac { b } { 2 x - 1 }$, find the values of $a$ and $b$.

AQA C4 2009 June Q1

5 marks Moderate -0.8

Use the Remainder Theorem to find the remainder when $3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5$ is divided by $3 x - 1$.
Express $\frac { 3 x ^ { 3 } + 8 x ^ { 2 } - 3 x - 5 } { 3 x - 1 }$ in the form $a x ^ { 2 } + b x + \frac { c } { 3 x - 1 }$, where $a , b$ and $c$ are integers.

AQA Paper 1 2024 June Q3

1 marks Easy -1.2

The expression $$\frac{12x^2 + 3x + 7}{3x - 5}$$ can be written as $$Ax + B + \frac{C}{3x - 5}$$ State the value of $A$ Circle your answer. [1 mark] $$3 \quad 4 \quad 7 \quad 9$$