Find constants using remainder theorem

7 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x + 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by ( \(x - 3\) ) the remainder is 30 , and that when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ) the remainder is 18 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, verify that ( \(x - 2\) ) is a factor of \(\mathrm { p } ( x )\) and hence factorise \(\mathrm { p } ( x )\) completely.

5 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + 9 x + 9\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that when \(\mathrm { p } ( x )\) is divided by \(( 2 x - 1 )\) the remainder is 10 .

1. Find the value of \(a\) and hence verify that ( \(x - 3\) ) is a factor of \(\mathrm { p } ( x )\).
2. When \(a\) has this value, solve the equation \(\mathrm { p } ( x ) = 0\).

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + a x + 4$$ where \(a\) is a constant.

1. Use the factor theorem to show that ( \(x + 1\) ) is a factor of \(\mathrm { p } ( x )\) for all values of \(a\).
2. Given that the remainder is - 42 when \(\mathrm { p } ( x )\) is divided by ( \(x - 2\) ), find the value of \(a\).
3. When \(a\) has the value found in part (ii), factorise \(\mathrm { p } \left( x ^ { 2 } \right)\) completely.

4. $$f ( x ) = ( x - 2 ) \left( 2 x ^ { 2 } + 5 x + k \right) + 21$$ where \(k\) is a constant.

1. State the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x - 2\) ) Given that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\)
2. show that \(k = 11\)
3. Hence
   1. fully factorise \(\mathrm { f } ( x )\),
   2. find the number of real solutions of the equation $$\mathrm { f } ( x ) = 0$$ giving a reason for your answer.

4. \(\mathrm { f } ( x ) = ( x - 3 ) \left( 3 x ^ { 2 } + x + a \right) - 35\) where \(a\) is a constant

1. State the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\). Given \(( 3 x - 2 )\) is a factor of \(\mathrm { f } ( x )\),
2. show that \(a = - 17\)
3. Using algebra and showing each step of your working, fully factorise \(\mathrm { f } ( x )\).

5 The cubic polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + a x + b$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\) and that the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\) is 16 .

1. Find the values of \(a\) and \(b\).
2. Hence verify that \(\mathrm { f } ( 2 ) = 0\), and factorise \(\mathrm { f } ( x )\) completely.

3

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + d\), where \(d\) is a constant. When \(\mathrm { f } ( x )\) is divided by ( \(2 x + 1\) ), the remainder is - 2 . Use the Remainder Theorem to find the value of \(d\).
2. The polynomial \(\mathrm { g } ( x )\) is defined by \(\mathrm { g } ( x ) = 8 x ^ { 3 } - 12 x ^ { 2 } - 2 x + 3\).
   1. Given that \(x = - \frac { 1 } { 2 }\) is a solution of the equation \(\mathrm { g } ( x ) = 0\), write \(\mathrm { g } ( x )\) as a product of three linear factors.
   2. The function h is defined by \(\mathrm { h } ( x ) = \frac { 4 x ^ { 2 } - 1 } { \mathrm {~g} ( x ) }\) for \(x > 2\). Simplify \(\mathrm { h } ( x )\), and hence show that h is a decreasing function.
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