Verify factor then simplify rational expression

Questions that verify a factor, factorise completely, then use the factorisation to simplify a rational expression or algebraic fraction.

7 questions · Moderate -0.5

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division
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Edexcel C3 Q5
12 marks Standard +0.3
5. (a) Show that \(( 2 x + 3 )\) is a factor of \(\left( 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)\).
(b) Hence, simplify $$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$ (c) Find the coordinates of the stationary points of the curve with equation $$y = \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$
AQA C4 2011 January Q2
10 marks Moderate -0.3
2
  1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 9 x ^ { 3 } + 18 x ^ { 2 } - x - 2\).
    1. Use the Factor Theorem to show that \(3 x + 1\) is a factor of \(\mathrm { f } ( x )\).
    2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
    3. Simplify \(\frac { 9 x ^ { 3 } + 21 x ^ { 2 } + 6 x } { \mathrm { f } ( x ) }\).
  2. When the polynomial \(9 x ^ { 3 } + p x ^ { 2 } - x - 2\) is divided by \(3 x - 2\), the remainder is - 4 . Find the value of the constant \(p\).
AQA C4 2011 June Q1
7 marks Moderate -0.8
1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } - 13 x + 6\).
  1. Find \(\mathrm { f } ( - 2 )\).
  2. Use the Factor Theorem to show that \(2 x - 3\) is a factor of \(\mathrm { f } ( x )\).
  3. Simplify \(\frac { 2 x ^ { 2 } + x - 6 } { \mathrm { f } ( x ) }\).
AQA C4 2008 June Q1
9 marks Moderate -0.3
1 The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 27 x ^ { 3 } - 9 x + 2\).
  1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(3 x + 1\).
    1. Show that f \(\left( - \frac { 2 } { 3 } \right) = 0\).
    2. Express \(\mathrm { f } ( x )\) as a product of three linear factors.
    3. Simplify $$\frac { 27 x ^ { 3 } - 9 x + 2 } { 9 x ^ { 2 } + 3 x - 2 }$$
OCR MEI C1 Q3
3 marks Moderate -0.8
Factorise and hence simplify \(\frac{3x^2 - 7x + 4}{x^2 - 1}\). [3]
OCR MEI C1 Q2
5 marks Easy -1.3
  1. Simplify \(3a^3b \times 4(ab)^2\). [2]
  2. Factorise \(x^2 - 4\) and \(x^2 - 5x + 6\). Hence express \(\frac{x^2 - 4}{x^2 - 5x + 6}\) as a fraction in its simplest form. [3]
AQA Paper 1 Specimen Q4
6 marks Moderate -0.3
\(p(x) = 2x^3 + 7x^2 + 2x - 3\)
  1. Use the factor theorem to prove that \(x + 3\) is a factor of \(p(x)\) [2 marks]
  2. Simplify the expression \(\frac{2x^3 + 7x^2 + 2x - 3}{4x^2 - 1}\), \(x \neq \pm \frac{1}{2}\) [4 marks]