Sketch curve using polynomial roots

A question is this type if and only if you must sketch y = p(x) after finding roots and possibly turning points of the polynomial.

4 questions · Moderate -0.8

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AQA C1 2013 January Q5
10 marks Moderate -0.8
5 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18$$
  1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x + 1\).
    1. Use the Factor Theorem to show that \(x - 3\) is a factor of \(\mathrm { p } ( x )\).
    2. Express \(\mathrm { p } ( x )\) as a product of linear factors.
  3. Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 3 x + 18\), stating the values of \(x\) where the curve meets the \(x\)-axis.
AQA C1 2010 June Q3
12 marks Moderate -0.8
3 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + 7 x ^ { 2 } + 7 x - 15$$
    1. Use the Factor Theorem to show that \(x + 3\) is a factor of \(\mathrm { p } ( x )\).
    2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 2\).
    1. Verify that \(\mathrm { p } ( - 1 ) < \mathrm { p } ( 0 )\).
    2. Sketch the curve with equation \(y = x ^ { 3 } + 7 x ^ { 2 } + 7 x - 15\), indicating the values where the curve crosses the coordinate axes.
AQA C1 2007 January Q1
11 marks Moderate -0.8
1 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 7 x + k$$ where \(k\) is a constant.
    1. Given that \(x + 2\) is a factor of \(\mathrm { p } ( x )\), show that \(k = 10\).
    2. Express \(\mathrm { p } ( x )\) as the product of three linear factors.
  2. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 3\).
  3. Sketch the curve with equation \(y = x ^ { 3 } - 4 x ^ { 2 } - 7 x + 10\), indicating the values where the curve crosses the \(x\)-axis and the \(y\)-axis. (You are not required to find the coordinates of the stationary points.)
AQA C1 2008 June Q6
11 marks Moderate -0.8
6 The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } + x ^ { 2 } - 8 x - 12\).
  1. Use the Remainder Theorem to find the remainder when \(\mathrm { p } ( x )\) is divided by \(x - 1\).
    1. Use the Factor Theorem to show that \(x + 2\) is a factor of \(\mathrm { p } ( x )\).
    2. Express \(\mathrm { p } ( x )\) as the product of linear factors.
    1. The curve with equation \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\) passes through the point \(( 0 , k )\). State the value of \(k\).
    2. Sketch the graph of \(y = x ^ { 3 } + x ^ { 2 } - 8 x - 12\), indicating the values of \(x\) where the curve touches or crosses the \(x\)-axis.