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

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1. It is given that \(a\) and \(b\) are positive constants. By sketching graphs of $$y = x ^ { 5 } \quad \text { and } \quad y = a - b x$$ on the same diagram, show that the equation $$x ^ { 5 } + b x - a = 0$$ has exactly one real root.
2. Use the iterative formula \(x _ { n + 1 } = \sqrt [ 5 ] { 53 - 2 x _ { n } }\), with a suitable starting value, to find the real root of the equation \(x ^ { 5 } + 2 x - 53 = 0\). Show the result of each iteration, and give the root correct to 3 decimal places.

2 The sequence defined by $$x _ { 1 } = 3 , \quad x _ { n + 1 } = \sqrt [ 3 ] { 31 - \frac { 5 } { 2 } x _ { n } }$$ converges to the number \(\alpha\).

1. Find the value of \(\alpha\) correct to 3 decimal places, showing the result of each iteration.
2. Find an equation of the form \(a x ^ { 3 } + b x + c = 0\), where \(a\), \(b\) and \(c\) are integers, which has \(\alpha\) as a root.

3 The sequence defined by the iterative formula $$x _ { n + 1 } = \sqrt [ 3 ] { } \left( 17 - 5 x _ { n } \right)$$ with \(x _ { 1 } = 2\), converges to \(\alpha\).

1. Use the iterative formula to find \(\alpha\) correct to 2 decimal places. You should show the result of each iteration.
2. Find a cubic equation of the form $$x ^ { 3 } + c x + d = 0$$ which has \(\alpha\) as a root.
3. Does this cubic equation have any other real roots? Justify your answer.

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1. Find the quotient and the remainder when \(3 x ^ { 3 } - 2 x ^ { 2 } + x + 7\) is divided by \(x ^ { 2 } - 2 x + 5\).
2. Hence, or otherwise, determine the values of the constants \(a\) and \(b\) such that, when \(3 x ^ { 3 } - 2 x ^ { 2 } + a x + b\) is divided by \(x ^ { 2 } - 2 x + 5\), there is no remainder.

3 When \(x ^ { 4 } - 2 x ^ { 3 } - 7 x ^ { 2 } + 7 x + a\) is divided by \(x ^ { 2 } + 2 x - 1\), the quotient is \(x ^ { 2 } + b x + 2\) and the remainder is \(c x + 7\). Find the values of the constants \(a , b\) and \(c\).

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1. Simplify \(\frac { x ^ { 3 } - x ^ { 2 } - 3 x - 9 } { x - 3 }\).
2. Hence or otherwise solve the equation \(x ^ { 3 } - x ^ { 2 } - 3 x - 9 = 6 ( x - 3 )\).

4.

1. Express $$\frac { 4 x } { x ^ { 2 } - 9 } - \frac { 2 } { x + 3 }$$ as a single fraction in its simplest form.
2. Simplify $$\frac { x ^ { 3 } - 8 } { 3 x ^ { 2 } - 8 x + 4 }$$

7.

1. Show that ( \(2 x + 3\) ) is a factor of ( \(\left. 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)\) and hence, simplify $$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$
2. Show that $$\int _ { 2 } ^ { 5 } \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } \mathrm {~d} x = \ln k$$ where \(k\) is an integer.

4 Find the values of \(A , B , C\) and \(D\) in the identity \(3 x ^ { 3 } - x ^ { 2 } + 2 \equiv A ( x - 1 ) ^ { 3 } + \left( x ^ { 3 } + B x ^ { 2 } + C x + D \right)\).

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = ( x + a ) ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are positive constants. The curve cuts the \(x\)-axis at \(P\) and has a maximum point at \(S\) and a minimum point at \(Q\).

1. Write down the coordinates of \(P\) and \(Q\) in terms of \(a\) and \(b\).
2. Show that \(G\), the area of the shaded region between the curve \(P S Q\) and the \(x\)-axis, is given by \(G = \frac { ( a + b ) ^ { 4 } } { 12 }\). The rectangle \(P Q R S T\) has \(R S T\) parallel to \(Q P\) and both \(P T\) and \(Q R\) are parallel to the \(y\)-axis.
3. Show that \(\frac { G } { \text { Area of } P Q R S T } = k\), where \(k\) is a constant independent of \(a\) and \(b\) and find the value of \(k\).

7. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C\) with equation \(y = x ^ { 4 } - 10 x ^ { 3 } + 33 x ^ { 2 } - 34 x\) and the line \(L\) with equation \(y = m x + c\) . The line \(L\) touches \(C\) at the points \(P\) and \(Q\) with \(x\) coordinates \(p\) and \(q\) respectively.

1. Explain why $$x ^ { 4 } - 10 x ^ { 3 } + 33 x ^ { 2 } - ( 34 + m ) x - c = ( x - p ) ^ { 2 } ( x - q ) ^ { 2 }$$ The finite region \(R\) ,shown shaded in Figure 3,is bounded by \(C\) and \(L\) .
2. Use integration by parts to show that the area of \(R\) is \(\frac { ( q - p ) ^ { 5 } } { 30 }\)
3. Show that $$( x - p ) ^ { 2 } ( x - q ) ^ { 2 } = x ^ { 4 } - 2 ( p + q ) x ^ { 3 } + S x ^ { 2 } - T x + U$$ where \(S , T\) and \(U\) are expressions to be found in terms of \(p\) and \(q\) .
4. Using part(a)and part(c)find the value of \(p\) ,the value of \(q\) and the equation of \(L\) .

2 Given that $$( x - p ) \left( 2 x ^ { 2 } + 9 x + 10 \right) = \left( x ^ { 2 } - 4 \right) ( 2 x + q )$$ for all values of \(x\), find the constants \(p\) and \(q\).

4 Solve the simultaneous equations $$4 x ^ { 2 } + y ^ { 2 } = 10 , \quad 2 x - y = 4$$

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1. Expand \(( x - 2 ) ^ { 2 } ( x + 1 )\), simplifying your answer.
2. Sketch the curve \(y = ( x - 2 ) ^ { 2 } ( x + 1 )\), indicating the coordinates of all intercepts with the axes.

1 Simplify \(( x - 5 ) \left( x ^ { 2 } + 3 \right) - ( x + 4 ) ( x - 1 )\).

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1. Simplify \(( 2 x - 3 ) ^ { 2 } - 2 ( 3 - x ) ^ { 2 }\).
2. Find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 3 x ^ { 2 } - 3 x + 4 \right) \left( 5 - 2 x - x ^ { 3 } \right)\).

4 When \(x ^ { 3 } + k x + 5\) is divided by \(x - 2\), the remainder is 3 . Use the remainder theorem to find the value of \(k\).

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1. Simplify \(3 a ^ { 3 } b \times 4 ( a b ) ^ { 2 }\).
2. Factorise \(x ^ { 2 } - 4\) and \(x ^ { 2 } - 5 x + 6\). Hence express \(\frac { x ^ { 2 } - 4 } { x ^ { 2 } - 5 x + 6 }\) as a fraction in its simplest form.

10 Simplify \(\left( m ^ { 2 } + 1 \right) ^ { 2 } - \left( m ^ { 2 } - 1 \right) ^ { 2 }\), showing your method.
Hence, given the right-angled triangle in Fig. 10, express \(p\) in terms of \(m\), simplifying your answer. \begin{figure}[h] \end{figure}

13 \begin{figure}[h] \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 \(\mathrm { f } ( x - 3 ) = 0\).

6 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 \(a x ^ { 3 } + b x ^ { 2 } + c x + d\).

7 When \(x ^ { 3 } + 2 x ^ { 2 } + 5 x + k\) is divided by ( \(x + 3\) ), the remainder is 6 . Find the value of \(k\).

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1. 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 \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
2. 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.
3. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).

3 Expand and simplify \(( n + 2 ) ^ { 3 } - n ^ { 3 }\).