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

670 questions

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OCR MEI C1 Q7
5 marks Moderate -0.8
7 Show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x ) = x ^ { 3 } - x ^ { 2 } - 4 x + 4\).
Hence solve the equation \(x ^ { 3 } - x ^ { 2 } - 4 x + 4 = 0\).
OCR MEI C1 Q5
5 marks Moderate -0.8
5 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } - 7 x + 6\).
  1. Show that ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\).
  2. Solve the equation \(\mathrm { f } ( x ) = 0\).
OCR MEI C1 Q9
5 marks Easy -1.2
9
  1. Show that \(( x - 1 ) ( x - 2 ) ( x - 3 ) - \left( x ^ { 3 } - x ^ { 2 } + 11 x - 12 \right) = 6 - 5 x ^ { 2 }\).
  2. Solve the equation \(6 - 5 x ^ { 2 } = 0\).
OCR MEI C1 Q1
2 marks Easy -1.2
1 Simplify \(( 3 x - 1 ) \left( 2 x ^ { 2 } - 5 x + 3 \right)\).
OCR MEI C1 Q11
12 marks Moderate -0.8
11 In this question \(\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 4 x + k\).
  1. You are asked to find the values of \(k\) which satisfy the following conditions.
    (A) The graph of \(y = \mathrm { f } ( x )\) goes through the origin.
    (B) The graph of \(y = \mathrm { f } ( x )\) intersects with the \(y\) axis at ( \(0 , - 2\) ).
    (C) ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\).
    (D) The remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) is 5 .
    (E) The graph of \(y = \mathrm { f } ( x )\) is as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-3_373_788_2131_584}
  2. Find the solution of the equation \(\mathrm { f } ( x ) = 0\) when \(k = 8\). Sketch a graph of \(y = \mathrm { f } ( x )\) in this case.
OCR MEI C1 Q7
5 marks Moderate -0.3
7 The remainder when \(x ^ { 3 } - 2 x + 4\) is divided by ( \(x - 2\) ) is twice the remainder when \(x ^ { 2 } + x + k\) is divided by ( \(x + 1\) ).
Find the value of \(k\).
OCR MEI C1 Q11
12 marks Moderate -0.3
11 Fig. 11 shows the graph of \(y = a x ^ { 2 } + b x + c\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-4_572_1509_465_285} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Explain why a must be negative.
  2. State two factors of \(y = a x ^ { 2 } + b x + c\).
  3. Hence, or otherwise, find the values of \(a , b\) and \(c\). Another function is given by \(y = x ^ { 2 } - 4 x + 10\).
  4. Write this in completed square form.
  5. Explain why the graphs of these two functions never meet.
OCR MEI C1 Q12
12 marks Standard +0.3
12 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5 x - 12\).
  1. Show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Find the other factors of \(\mathrm { f } ( x )\).
  3. State the coordinates where the graph of \(y = \mathrm { f } ( x )\) cuts the \(x\) axis. Hence sketch the graph of \(y = \mathrm { f } ( x )\).
  4. On the same graph sketch also \(y = x ( x - 1 ) ( x - 2 )\) Label the two points of intersection of the two curves A and B .
  5. By equating the two curves, show that the \(x\) coordinates of A and B satisfy the equation \(3 x ^ { 2 } + x - 4 = 0\).
    Solve this equation to find the \(x\)-coordinates of A and B .
OCR C1 Q8
9 marks Moderate -0.3
8. $$f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12$$
  1. Show that $$( x + 1 ) ( x - 3 ) ( x - 4 ) \equiv x ^ { 3 } - 6 x ^ { 2 } + 5 x + 12 .$$
  2. Sketch the curve \(y = \mathrm { f } ( x )\), showing the coordinates of any points of intersection with the coordinate axes.
  3. Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
    1. \(\quad y = \mathrm { f } ( x + 3 )\),
    2. \(y = \mathrm { f } ( - x )\).
OCR C1 Q5
7 marks Moderate -0.5
  1. Given that
$$\left( x ^ { 2 } + 2 x - 3 \right) \left( 2 x ^ { 2 } + k x + 7 \right) \equiv 2 x ^ { 4 } + A x ^ { 3 } + A x ^ { 2 } + B x - 21 ,$$ find the values of the constants \(k , A\) and \(B\).
OCR C1 Q1
3 marks Moderate -0.8
  1. \(\quad \mathrm { f } ( x ) = ( \sqrt { x } + 3 ) ^ { 2 } + ( 1 - 3 \sqrt { x } ) ^ { 2 }\).
Show that \(\mathrm { f } ( x )\) can be written in the form \(a x + b\) where \(a\) and \(b\) are integers to be found.
OCR C1 Q2
4 marks Moderate -0.3
2. Find in exact form the real solutions of the equation $$x ^ { 4 } = 5 x ^ { 2 } + 14 .$$
OCR C1 Q4
5 marks Moderate -0.8
4. Express each of the following in the form \(p + q \sqrt { 2 }\) where \(p\) and \(q\) are rational.
  1. \(( 4 - 3 \sqrt { 2 } ) ^ { 2 }\)
  2. \(\frac { 1 } { 2 + \sqrt { 2 } }\)
OCR MEI C1 Q1
12 marks Moderate -0.8
1 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).
  1. Sketch the curve \(y = \mathrm { f } ( x )\).
  2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
  3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
  4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.
OCR MEI C1 Q2
12 marks Moderate -0.3
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a94593ca-d84d-4747-ac19-b05389e85b3c-1_781_1462_1118_342} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} Fig. 12 shows the graph of a cubic curve. It intersects the axes at \(( - 5,0 ) , ( - 2,0 ) , ( 1.5,0 )\) and \(( 0 , - 30 )\).
  1. Use the intersections with both axes to express the equation of the curve in a factorised form.
  2. Hence show that the equation of the curve may be written as \(y = 2 x ^ { 3 } + 11 x ^ { 2 } - x - 30\).
  3. Draw the line \(y = 5 x + 10\) accurately on the graph. The curve and this line intersect at ( \(- 2,0\) ); find graphically the \(x\)-coordinates of the other points of intersection.
  4. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2 x ^ { 2 } + 7 x - 20 = 0$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection.
OCR MEI C1 Q3
12 marks Moderate -0.8
3 You are given that \(\mathrm { f } ( x ) = ( 2 x - 3 ) ( x + 2 ) ( x + 4 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. State the roots of \(\mathrm { f } ( x - 2 ) = 0\).
  3. You are also given that \(\mathrm { g } ( x ) = \mathrm { f } ( x ) + 15\).
    (A) Show that \(\mathrm { g } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } - 2 x - 9\).
    (B) Show that \(\mathrm { g } ( 1 ) = 0\) and hence factorise \(\mathrm { g } ( x )\) completely.
OCR MEI C1 Q5
12 marks Moderate -0.8
5 A cubic curve has equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis where \(x = - , \frac { 1 } { 2 }\) and 5 .
  1. Write down three linear factors of \(\mathrm { f } ( x )\). Hence find the equation of the curve in the form \(y = 2 x ^ { 3 } + a x ^ { 2 } + b x + c\).
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 0 } { - 8 }\). State the coordinates of the point where the translated curve intersects the \(y\)-axis.
  4. The curve \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\) to give the curve \(y = \mathrm { g } ( x )\). Find an expression in factorised form for \(\mathrm { g } ( x )\) and state the coordinates of the point where the curve \(y = \mathrm { g } ( x )\) intersects the \(y\)-axis.
OCR MEI C1 Q2
13 marks Moderate -0.3
2 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 3 x ^ { 2 } - 23 x + 12\).
  1. Show that \(x = - 3\) is a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) fully.
  2. Sketch the curve \(y = \mathrm { f } ( x )\).
  3. Find the \(x\)-coordinates of the points where the line \(y = 4 x + 12\) intersects \(y = \mathrm { f } ( x )\).
OCR MEI C1 Q3
12 marks Moderate -0.3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6be6c0b0-76b7-49c0-bf1b-dc6f8f79981b-2_836_906_361_675} \captionsetup{labelformat=empty} \caption{Fig. 12}
\end{figure} Fig. 12 shows the graph of \(y = \frac { 4 } { x ^ { 2 } }\).
  1. On the copy of Fig. 12, draw accurately the line \(y = 2 x + 5\) and hence find graphically the three roots of the equation \(\frac { 4 } { x ^ { 2 } } = 2 x + 5\).
    [0pt] [3]
  2. Show that the equation you have solved in part (i) may be written as \(2 x ^ { 3 } + 5 x ^ { 2 } - 4 = 0\). Verify that \(x = - 2\) is a root of this equation and hence find, in exact form, the other two roots.
    [0pt] [6]
  3. By drawing a suitable line on the copy of Fig. 12, find the number of real roots of the equation \(x ^ { 3 } + 2 x ^ { 2 } - 4 = 0\).
  4. You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
    (A) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
  5. You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
    (A) Show that \(\mathrm { g } ( 5 ) = 0\).
    (B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
    (C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
  6. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).
OCR MEI C1 Q1
12 marks Moderate -0.3
1 You are given that \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } - x - 30\).
  1. Use the factor theorem to find a root of \(\mathrm { f } ( x ) = 0\) and hence factorise \(\mathrm { f } ( x )\) completely.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 1 } { 0 }\). Show that the equation of the translated graph may be written as $$y = x ^ { 3 } + 3 x ^ { 2 } - 10 x - 24$$
OCR MEI C1 Q2
5 marks Moderate -0.8
2 You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ^ { 2 } ( 2 x - 5 )\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. Express \(\mathrm { f } ( x )\) in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d\).
OCR MEI C1 Q3
13 marks Moderate -0.3
3
  1. You are given that \(\mathrm { f } ( x ) = ( x + 1 ) ( x - 2 ) ( x - 4 )\).
    (A) Show that \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8\).
    (B) Sketch the graph of \(y = \mathrm { f } ( x )\).
    (C) The graph of \(y = \mathrm { f } ( x )\) is translated by \(\binom { 3 } { 0 }\). 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.
  2. Show that 3 is a root of \(x ^ { 3 } - 5 x ^ { 2 } + 2 x + 8 = - 4\). Hence solve this equation completely, giving the other roots in surd form.
OCR MEI C1 Q4
12 marks Moderate -0.3
4 You are given that \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 7 x ^ { 2 } - 7 x - 12\).
  1. Verify that \(x = - 4\) is a root of \(\mathrm { f } ( x ) = 0\).
  2. Hence express \(\mathrm { f } ( x )\) in fully factorised form.
  3. Sketch the graph of \(y = \mathrm { f } ( x )\).
  4. Show that \(\mathrm { f } ( x - 4 ) = 2 x ^ { 3 } - 17 x ^ { 2 } + 33 x\).
OCR MEI C1 Q5
12 marks Moderate -0.3
5 A cubic polynomial is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\).
  1. Show that \(( x - 3 ) \left( 2 x ^ { 2 } + 5 x + 4 \right) = 2 x ^ { 3 } - x ^ { 2 } - 11 x - 12\). Hence show that \(\mathrm { f } ( x ) = 0\) has exactly one real root.
  2. Show that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = - 22\) and find the other roots of this equation.
  3. Using the results from the previous parts, sketch the graph of \(y = \mathrm { f } ( x )\).
OCR MEI C1 Q2
12 marks Moderate -0.8
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{618d118a-2557-42f3-9b55-4a55dda93a97-1_449_376_631_889} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure} Fig. 13 shows a sketch of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 5 x + 2\).
  1. Use the fact that \(x = 2\) is a root of \(\mathrm { f } ( x ) = 0\) to find the exact values of the other two roots of \(\mathrm { f } ( x ) = 0\), expressing your answers as simply as possible.
  2. Show that \(\mathrm { f } ( x - 3 ) = x ^ { 3 } - 9 x ^ { 2 } + 22 x - 10\).
  3. Write down the roots of \(\mathrm { f } ( x - 3 ) = 0\).