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

670 questions

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SPS SPS SM Pure 2023 September Q8
9 marks Moderate -0.3
A cubic curve \(C\) has equation $$y = (3-x)(4+x)^2.$$
  1. Sketch the graph of \(C\). [3] The sketch must include any points where the graph meets the coordinate axes.
  2. Sketch in separate diagrams the graph of \(\ldots\)
    1. \(\ldots y = (3-2x)(4+2x)^2\). [2]
    2. \(\ldots y = (3+x)(4-x)^2\). [2]
    3. \(\ldots y = (2-x)(5+x)^2\). [2]
    Each of the sketches must include any points where the graph meets the coordinate axes.
SPS SPS FM 2024 October Q9
6 marks Standard +0.8
  1. Factorise \(8xy - 4x + 6y - 3\) into the form \((ax + b)(cy + d)\) where \(a, b, c\) and \(d\) are integers
  2. Hence, or otherwise, solve $$8\sin(x^2)\cos\left(e^{\frac{x}{3}}\right) - 4\sin(x^2) + 6\cos\left(e^{\frac{x}{3}}\right) - 3 = 0$$ where \(0° < x < 19°\), giving your answers to 1 decimal place.
[6 marks]
SPS SPS SM 2023 October Q8
7 marks Standard +0.3
In this question you must show detailed reasoning. The curve \(C_1\) has equation \(y = 8 - 10x + 6x^2 - x^3\) The curve \(C_2\) has equation \(y = x^2 - 12x + 14\)
  1. Verify that when \(x = 1\) the curves \(C_1\) and \(C_2\) intersect. [2]
The curves also intersect when \(x = k\). Given that \(k < 0\)
  1. use algebra to find the exact value of \(k\). [5]
SPS SPS FM 2024 October Q5
9 marks Standard +0.3
In this question you must show detailed reasoning. The polynomial \(f(x)\) is given by $$f(x) = x^3 + 6x^2 + x - 4.$$
    1. Show that \((x + 1)\) is a factor of \(f(x)\). [1]
    2. Hence find the exact roots of the equation \(f(x) = 0\). [4]
    1. Show that the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ can be written in the form \(f(x) = 0\). [3]
    2. Explain why the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ has only one real root and state the exact value of this root. [1]
SPS SPS SM 2024 October Q9
9 marks Moderate -0.3
In this question you must show detailed reasoning. The polynomial f(x) is given by $$f(x) = x^3 + 6x^2 + x - 4.$$
    1. Show that \((x + 1)\) is a factor of f(x). [1]
    2. Hence find the exact roots of the equation f(x) = 0. [4]
    1. Show that the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ can be written in the form f(x) = 0. [3]
    2. Explain why the equation $$2\log_2(x + 3) + \log_2 x - \log_2(4x + 2) = 1$$ has only one real root and state the exact value of this root. [1]
SPS SPS FM 2025 February Q1
7 marks Challenging +1.2
The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). \includegraphics{figure_1} The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is 48 units\(^2\), determine the \(y\)-coordinate of \(P\). [7]
SPS SPS SM 2025 February Q1
2 marks Easy -1.2
Given that \((x - 2)\) is a factor of \(2x^3 + kx - 4\), find the value of the constant \(k\). [2]
SPS SPS SM 2025 February Q4
6 marks Moderate -0.3
  1. The number \(K\) is defined by \(K = n^3 + 1\), where \(n\) is an integer greater than \(2\). Given that \(n^3 + 1 = (n + 1) (n^2 + bn + c)\), find the constants \(b\) and \(c\). [1]
  2. Prove that \(K\) has at least two distinct factors other than \(1\) and \(K\). [5]
SPS SPS SM 2025 February Q8
7 marks Challenging +1.2
\includegraphics{figure_8} The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is \(48\) units\(^2\), determine the \(y\)-coordinate of \(P\). [7]
SPS SPS FM 2025 October Q4
8 marks Moderate -0.3
The cubic polynomial \(2x^3 - kx^2 + 4x + k\), where \(k\) is a constant, is denoted by f(x). It is given that f'(2) = 16.
  1. Show that \(k = 3\). [3]
For the remainder of the question, you should use this value of \(k\).
  1. Use the factor theorem to show that \((2x + 1)\) is a factor of f(x). [2]
  2. Hence show that the equation f(x) = 0 has only one real root. [3]
SPS SPS SM 2025 October Q10
7 marks Moderate -0.3
\(f(x) = x^4 + bx + c\) \((x-2)\) is a factor of \(f(x)\). \(f(-3) = 35\).
  1. Find \(b\) and \(c\). [4]
  2. Hence express \(f(x)\) as the product of linear and cubic factors. [3]
SPS SPS FM 2026 November Q2
7 marks Easy -1.2
\(f(x) = x^3 + 4x^2 + x - 6\).
  1. Use the factor theorem to show that \((x + 2)\) is a factor of \(f(x)\). [2]
  2. Factorise \(f(x)\) completely. [4]
  3. Write down all the solutions to the equation $$x^3 + 4x^2 + x - 6 = 0.$$ [1]
OCR Further Additional Pure 2018 September Q1
5 marks Standard +0.8
  1. Write the number \(100011_n\), where \(n \geq 2\), as a polynomial in \(n\). [1]
  2. Show that \(n^2 + n + 1\) is a factor of this expression. [2]
  3. Hence show that \(100011_n\) is composite in any number base \(n \geq 2\). [2]
OCR H240/02 2018 December Q2
5 marks Moderate -0.3
\(\text{f}(x)\) is a cubic polynomial in which the coefficient of \(x^3\) is 1. The equation \(\text{f}(x) = 0\) has exactly two roots.
  1. Sketch a possible graph of \(y = \text{f}(x)\). [2]
It is now given that the two roots are \(x = 2\) and \(x = 3\).
  1. Find, in expanded form, the two possible polynomials \(\text{f}(x)\). [3]
OCR H240/02 2018 December Q7
5 marks Standard +0.8
  1. Show that, if \(n\) is a positive integer, then \((x^n - 1)\) is divisible by \((x - 1)\). [1]
  2. Hence show that, if \(k\) is a positive integer, then \(2^{8k} - 1\) is divisible by 17. [4]
Pre-U Pre-U 9794/2 2010 June Q6
10 marks Standard +0.3
  1. Express \(\frac{x-1}{x^2+2x+1}\) in the form \(\frac{A}{x+1} + \frac{B}{(x+1)^2}\), where \(A\) and \(B\) are integers. [2]
  2. Find the quotient and remainder when \(2y^2 + 1\) is divided by \(y + 1\). [2]
  3. A curve in the \(x\)-\(y\) plane passes through the point \((0, 2)\) and satisfies the differential equation $$(2y^2 + 1)(x^2 + 2x + 1)\frac{dy}{dx} = (x - 1)(y + 1).$$ By solving the differential equation find the equation of the curve in implicit form. [6]
Pre-U Pre-U 9794/2 2011 June Q1
5 marks Easy -1.3
  1. Show that \(x = 4\) is a root of \(x^3 - 12x - 16 = 0\). [2]
  2. Hence completely factorise the expression \(x^3 - 12x - 16\). [3]
Pre-U Pre-U 9794/2 2016 June Q1
3 marks Easy -1.3
  1. Find the remainder when \(x^3 + 2x\) is divided by \(x + 2\). [2]
  2. Write down the value of \(k\) for which \(x + 2\) is a factor of \(x^3 + 2x + k\). [1]
Pre-U Pre-U 9794/2 Specimen Q5
9 marks Standard +0.3
  1. Divide the quartic \(2x^4 - 5x^3 + 4x^2 + 2x - 3\) by the quadratic \(x^2 + x - 2\), identifying the quotient and the remainder. [4]
    1. Show that \((x - 1)\) is a factor of \(nx^{n+1} - (n + 1)x^n + 1\), where \(n\) is a positive integer. [1]
    2. Hence, or otherwise, find all the roots of \(3x^4 - 4x^3 + 1 = 0\). [4]
Edexcel AEA 2015 June Q2
9 marks Challenging +1.8
  1. Show that \((x + 1)\) is a factor of \(2x^3 + 3x^2 - 1\) [1]
  2. Solve the equation $$\sqrt{x^2 + 2x + 5} = x + \sqrt{2x + 3}$$ [8]