1.02j Manipulate polynomials: expanding, factorising, division, factor theorem

670 questions

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Edexcel C2 Q3
6 marks Moderate -0.8
  1. Use the factor theorem to show that \((x + 4)\) is a factor of \(2x^3 + x^2 - 25x + 12\). [2]
  2. Factorise \(2x^3 + x^2 - 25x + 12\) completely. [4]
Edexcel C2 Q1
8 marks Moderate -0.8
\(f(x) = 2x^3 + x^2 - 5x + c\), where \(c\) is a constant. Given that \(f(1) = 0\),
  1. find the value of \(c\), [2]
  2. factorise \(f(x)\) completely, [4]
  3. find the remainder when \(f(x)\) is divided by \((2x - 3)\). [2]
Edexcel C2 Q2
6 marks Moderate -0.8
\(f(x) = 3x^3 - 5x^2 - 16x + 12\).
  1. Find the remainder when \(f(x)\) is divided by \((x - 2)\). [2]
Given that \((x + 2)\) is a factor of \(f(x)\),
  1. factorise \(f(x)\) completely. [4]
Edexcel C2 Q1
7 marks Moderate -0.8
  1. Find the remainder when \(x^3 - 2x^2 - 4x + 8\) is divided by
    1. \(x - 3\),
    2. \(x + 2\). [3]
  2. Hence, or otherwise, find all the solutions to the equation \(x^3 - 2x^2 - 4x + 8 = 0\). [4]
Edexcel C2 2008 January Q1
7 marks Moderate -0.8
  1. Find the remainder when $$x^3 - 2x^2 - 4x + 8$$ is divided by
    1. \(x - 3\),
    2. \(x + 2\).
    [3]
  2. Hence, or otherwise, find all the solutions to the equation $$x^3 - 2x^2 - 4x + 8 = 0.$$ [4]
Edexcel C2 Q1
4 marks Moderate -0.8
\(f(x) = 2x^3 - x^2 + px + 6\), where \(p\) is a constant. Given that \((x - 1)\) is a factor of \(f(x)\), find
  1. the value of \(p\), [2]
  2. the remainder when \(f(x)\) is divided by \((2x + 1)\). [2]
Edexcel C2 Q12
6 marks Moderate -0.3
$$f(x) = ax^3 + bx^2 - 7x + 14, \text{ where } a \text{ and } b \text{ are constants.}$$ Given that when \(f(x)\) is divided by \((x - 1)\) the remainder is 9,
  1. write down an equation connecting \(a\) and \(b\). [2]
Given also that \((x + 2)\) is a factor of \(f(x)\),
  1. Find the values of \(a\) and \(b\). [4]
Edexcel C2 Q13
7 marks Moderate -0.8
$$f(x) = x^3 - x^2 - 7x + c, \text{ where } c \text{ is a constant.}$$ Given that \(f(4) = 0\),
  1. Find the value of \(c\), [2]
  2. factorise \(f(x)\) as the product of a linear factor and a quadratic factor. [3]
  3. Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(f(x) = 0\). [2]
Edexcel C2 Q20
3 marks Moderate -0.8
$$f(x) = 4x^3 + 3x^2 - 2x - 6.$$ Find the remainder when \(f(x)\) is divided by \((2x + 1)\). [3]
Edexcel C2 Q29
7 marks Moderate -0.3
$$f(x) = x^3 + ax^2 + bx - 10, \text{ where } a \text{ and } b \text{ are constants.}$$ When \(f(x)\) is divided by \((x - 3)\), the remainder is 14. When \(f(x)\) is divided by \((x + 1)\), the remainder is \(-18\).
  1. Find the value of \(a\) and the value of \(b\). [5]
  2. Show that \((x - 2)\) is a factor of \(f(x)\). [2]
Edexcel C2 Q30
6 marks Moderate -0.8
  1. Using the factor theorem, show that \((x + 3)\) is a factor of $$x^3 - 3x^2 - 10x + 24.$$ [2]
  2. Factorise \(x^3 - 3x^2 - 10x + 24\) completely. [4]
Edexcel FP1 Q1
5 marks Moderate -0.3
$$\text{f}(x) = 2x^3 - 8x^2 + 7x - 3.$$ Given that \(x = 3\) is a solution of the equation f\((x) = 0\), solve f\((x) = 0\) completely. [5]
Edexcel M2 2014 January Q9
12 marks Moderate -0.3
A curve with equation \(y = f(x)\) passes through the point \((3, 6)\). Given that $$f'(x) = (x - 2)(3x + 4)$$
  1. use integration to find \(f(x)\). Give your answer as a polynomial in its simplest form. [5]
  2. Show that \(f(x) = (x - 2)^2(x + p)\), where \(p\) is a positive constant. State the value of \(p\). [3]
  3. Sketch the graph of \(y = f(x)\), showing the coordinates of any points where the curve touches or crosses the coordinate axes. [4]
Edexcel C1 Q2
7 marks Standard +0.3
Solve the simultaneous equations $$x - 3y + 1 = 0,$$ $$x^2 - 3xy + y^2 = 11.$$ [7]
OCR C1 2013 January Q5
6 marks Easy -1.3
  1. Simplify \((x + 4)(5x - 3) - 3(x - 2)^2\). [3]
  2. The coefficient of \(x^2\) in the expansion of $$(x + 3)(x + k)(2x - 5)$$ is \(-3\). Find the value of the constant \(k\). [3]
OCR C1 2006 June Q4
8 marks Easy -1.2
  1. By expanding the brackets, show that $$(x - 4)(x - 3)(x + 1) = x^3 - 6x^2 + 5x + 12.$$ [3]
  2. Sketch the curve $$y = x^3 - 6x^2 + 5x + 12,$$ giving the coordinates of the points where the curve meets the axes. Label the curve \(C_1\). [3]
  3. On the same diagram as in part (ii), sketch the curve $$y = -x^3 + 6x^2 - 5x - 12.$$ Label this curve \(C_2\). [2]
OCR C1 2013 June Q10
14 marks Standard +0.3
The curve \(y = (1 - x)(x^2 + 4x + k)\) has a stationary point when \(x = -3\).
  1. Find the value of the constant \(k\). [7]
  2. Determine whether the stationary point is a maximum or minimum point. [2]
  3. Given that \(y = 9x - 9\) is the equation of the tangent to the curve at the point \(A\), find the coordinates of \(A\). [5]
OCR MEI C1 Q11
12 marks Moderate -0.3
A cubic polynomial is given by \(f(x) = x^3 + x^2 - 10x + 8\).
  1. Show that \((x - 1)\) is a factor of \(f(x)\). Factorise \(f(x)\) fully. Sketch the graph of \(y = f(x)\). [7]
  2. The graph of \(y = f(x)\) is translated by \(\begin{pmatrix} -3 \\ 0 \end{pmatrix}\). Write down an equation for the resulting graph. You need not simplify your answer. Find also the intercept on the \(y\)-axis of the resulting graph. [5]
OCR MEI C1 2006 January Q6
3 marks Easy -1.2
When \(x^3 + 3x + k\) is divided by \(x - 1\), the remainder is 6. Find the value of \(k\). [3]
OCR MEI C1 2006 January Q12
13 marks Standard +0.3
  1. Sketch the graph of \(y = x(x - 3)^2\). [3]
  2. Show that the equation \(x(x - 3)^2 = 2\) can be expressed as \(x^3 - 6x^2 + 9x - 2 = 0\). [2]
  3. Show that \(x = 2\) is one root of this equation and find the other two roots, expressing your answers in surd form. Show the location of these roots on your sketch graph in part (i). [8]
OCR MEI C1 2006 June Q2
2 marks Moderate -0.8
One root of the equation \(x^3 + ax^2 + 7 = 0\) is \(x = -2\). Find the value of \(a\). [2]
OCR MEI C1 2006 June Q12
12 marks Moderate -0.8
You are given that \(\text{f}(x) = x^3 + 9x^2 + 20x + 12\).
  1. Show that \(x = -2\) is a root of \(\text{f}(x) = 0\). [2]
  2. Divide \(\text{f}(x)\) by \(x + 6\). [2]
  3. Express \(\text{f}(x)\) in fully factorised form. [2]
  4. Sketch the graph of \(y = \text{f}(x)\). [3]
  5. Solve the equation \(\text{f}(x) = 12\). [3]
OCR MEI C1 2009 June Q3
3 marks Moderate -0.8
When \(x^3 - kx + 4\) is divided by \(x - 3\), the remainder is 1. Use the remainder theorem to find the value of \(k\). [3]
OCR MEI C1 2009 June Q12
13 marks Moderate -0.8
  1. You are given that \(\text{f}(x) = (x + 1)(x - 2)(x - 4)\).
    1. Show that \(\text{f}(x) = x^3 - 5x^2 + 2x + 8\). [2]
    2. Sketch the graph of \(y = \text{f}(x)\). [3]
    3. The graph of \(y = \text{f}(x)\) is translated by \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\). State an equation for the resulting graph. You need not simplify your answer. Find the coordinates of the point at which the resulting graph crosses the \(y\)-axis. [3]
  2. Show that 3 is a root of \(x^3 - 5x^2 + 2x + 8 = -4\). Hence solve this equation completely, giving the other roots in surd form. [5]
OCR MEI C1 2010 June Q6
5 marks Moderate -0.3
You are given that • the coefficient of \(x^3\) in the expansion of \((5 + 2x^2)(x^3 + kx + m)\) is 29, • when \(x^3 + kx + m\) is divided by \((x - 3)\), the remainder is 59. Find the values of \(k\) and \(m\). [5]