Single unknown constant

1

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 \(( x + 3 )\).

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 \(k\).

One root of the equation \(x^3 + ax^2 + 7 = 0\) is \(x = -2\). Find the value of \(a\). [2]

One root of the equation \(x^3 + ax^2 + 7 = 0\) is \(x = -2\). Find the value of \(a\). [2]

\(p(x) = x^3 - 5x^2 + 3x + a\), where \(a\) is a constant. Given that \(x - 3\) is a factor of \(p(x)\), find the value of \(a\) Circle your answer. [1 mark] \(-9\) \quad\quad \(-3\) \quad\quad \(3\) \quad\quad \(9\)

Given that \((x - 2)\) is a factor of \(2x^3 + kx - 4\), find the value of the constant \(k\). [2]