Multiple unknowns with derivative condition

Given a polynomial with unknown constants where at least one condition involves the derivative being zero or having a specific factor, requiring differentiation before applying factor/remainder theorem.

7 questions · Standard +0.3

1.02j Manipulate polynomials: expanding, factorising, division, factor theorem
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CAIE P3 2009 November Q5
8 marks Standard +0.3
5 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x - 4\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). The result of differentiating \(\mathrm { p } ( x )\) with respect to \(x\) is denoted by \(\mathrm { p } ^ { \prime } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and of \(\mathrm { p } ^ { \prime } ( x )\).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
CAIE P3 2022 June Q5
8 marks Standard +0.3
5 The polynomial \(a x ^ { 3 } - 10 x ^ { 2 } + b x + 8\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 2 )\) is a factor of both \(\mathrm { p } ( x )\) and \(\mathrm { p } ^ { \prime } ( x )\).
  1. Find the values of \(a\) and \(b\).
  2. When \(a\) and \(b\) have these values, factorise \(\mathrm { p } ( x )\) completely.
Edexcel PMT Mocks Q7
9 marks Standard +0.3
7. A curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given that
  • \(\mathrm { f } ^ { \prime } ( x ) = 18 x ^ { 2 } + 2 a x + b\)
  • the \(y\)-intercept of \(C\) is - 48
  • the point \(A\), with coordinates \(( - 1,45 )\) lies on \(C\) a. Show that \(a - b = 99\) b. Find the value of \(a\) and the value of \(b\).
    c. Show that \(( 2 x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
CAIE P3 2014 June Q5
7 marks Standard +0.8
  1. The polynomial \(f(x)\) is of the form \((x - 2)^2g(x)\), where \(g(x)\) is another polynomial. Show that \((x - 2)\) is a factor of \(f'(x)\). [2]
  2. The polynomial \(x^5 + ax^4 + 3x^3 + bx^2 + a\), where \(a\) and \(b\) are constants, has a factor \((x - 2)^2\). Using the factor theorem and the result of part (i), or otherwise, find the values of \(a\) and \(b\). [5]
Edexcel C1 Q4
5 marks Standard +0.3
\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \((-1, 0)\) and touches the \(x\)-axis at the point \((3, 0)\). Show that \(a = -5\) and find the values of \(b\) and \(c\). [5]
OCR C1 Q3
5 marks Standard +0.3
\includegraphics{figure_3} The diagram shows the curve with equation \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \((-1, 0)\) and touches the \(x\)-axis at the point \((3, 0)\). Show that \(a = -5\) and find the values of \(b\) and \(c\). [5]
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]