Single unknown constant

1

1. Tb p lm ial \(2 x ^ { 3 } + a x ^ { 2 } - a x - 2\) wh re \(a\) is a co tan, is d h ed \(\mathrm { y } \quad \mathrm { p } x\) ). It is g n th t \(( x + 1\) is a facto \(6 \quad ( x )\). Fid b le \(6 a\).
2. Wh n \(a \mathbf { b }\) s th s le , f in e remaid r wh \(\underline { \mathrm { p } } \quad x )\) is \(\dot { \mathbf { d } } \dot { \mathbf { v } } \mathbf { d }\) dt \(\quad x + \beta\).

3

1. The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - a x - 12\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\). Find the value of \(a\).
2. When \(a\) has this value, find the remainder when \(\mathrm { p } ( x )\) is divided by \(( x + 3 )\).

2 The polynomial \(x ^ { 3 } - 2 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, find the quadratic factor of \(\mathrm { p } ( x )\).

4 You are given that \(\mathrm { f } ( x ) = x ^ { 4 } + a x - 6\) and that \(x - 2\) is a factor of \(\mathrm { f } ( x )\).
Find the value of \(a\).

1 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } - a x - 14\), where \(a\) is a constant.

1. Given that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. Using this value of \(a\), find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 1\) ).

1. $$g ( x ) = 3 x ^ { 3 } - 20 x ^ { 2 } + ( k + 17 ) x + k$$ where \(k\) is a constant.
Given that \(( x - 3 )\) is a factor of \(\mathrm { g } ( x )\), find the value of \(k\).

1. $$f ( x ) = a x ^ { 3 } + 10 x ^ { 2 } - 3 a x - 4$$ Given that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\), find the value of the constant \(a\).
You must make your method clear.

1. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + a x + a$$ Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of the constant \(a\).

1 Given that \(( x - 2 )\) is a factor of \(2 x ^ { 3 } + k x - 4\), find the value of the constant \(k\).

1. Given that \(( x - 2 )\) is a factor of \(2 x ^ { 3 } + k x - 4\), find the value of the constant \(k\).
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2. (a)
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The diagram shows a model for the roof of a toy building. The roof is in the form of a solid triangular prism \(A B C D E F\). The base \(A C F D\) of the roof is a horizontal rectangle, and the cross-section \(A B C\) of the roof is an isosceles triangle with \(A B = B C\). The lengths of \(A C\) and \(C F\) are \(2 x \mathrm {~cm}\) and \(y \mathrm {~cm}\) respectively, and the height of \(B E\) above the base of the roof is \(x \mathrm {~cm}\). The total surface area of the five faces of the roof is \(600 \mathrm {~cm} ^ { 2 }\) and the volume of the roof is \(V \mathrm {~cm} ^ { 3 }\). Show that \(V = k x \left( 300 - x ^ { 2 } \right)\), where \(k = \sqrt { a } + b\) and \(\alpha\) and \(b\) are integers to be determined.
(b) Use differentiation to determine the value of \(x\) for which the volume of the roof is a maximum.
(c) Find the maximum volume of the roof. Give your answer in \(\mathrm { cm } ^ { 3 }\), correct to the nearest integer.
(d) Explain why, for this roof, \(x\) must be less than a certain value, which you should state.
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2 One root of the equation \(x ^ { 3 } + a x ^ { 2 } + 7 = 0\) is \(x = - 2\). Find the value of \(a\).

1 \(\mathrm { p } ( x ) = x ^ { 3 } - 5 x ^ { 2 } + 3 x + a\), where \(a\) is a constant.
Given that \(x - 3\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\)
Circle your answer.
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\(- 9 - 339\)