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

3 The polynomial \(x ^ { 4 } + 4 x ^ { 2 } + x + a\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x ^ { 2 } + x + 2\) ) is a factor of \(\mathrm { p } ( x )\).
Find the value of \(a\) and the other quadratic factor of \(p ( x )\).

4 The polynomial \(x ^ { 4 } - 2 x ^ { 3 } - 2 x ^ { 2 } + a\) is denoted by \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ( x )\) is divisible by \(x ^ { 2 } - 4 x + 4\).

1. Find the value of \(a\).
2. When \(a\) has this value, show that \(\mathrm { f } ( x )\) is never negative.

5 The polynomial \(x ^ { 4 } + 5 x + a\) is denoted by \(\mathrm { p } ( x )\). It is given that \(x ^ { 2 } - x + 3\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and factorise \(\mathrm { p } ( x )\) completely.
2. Hence state the number of real roots of the equation \(\mathrm { p } ( x ) = 0\), justifying your answer.

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 \(\mathrm { p } ( x )\).

5 The polynomial \(2 x ^ { 3 } + 5 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is 9 .

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

4 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = 12 x ^ { 3 } + 25 x ^ { 2 } - 4 x - 12$$

1. Show that \(\mathrm { f } ( - 2 ) = 0\) and factorise \(\mathrm { f } ( x )\) completely.
2. Given that $$12 \times 27 ^ { y } + 25 \times 9 ^ { y } - 4 \times 3 ^ { y } - 12 = 0$$ state the value of \(3 ^ { y }\) and hence find \(y\) correct to 3 significant figures.

5 The polynomial \(a x ^ { 3 } + b x ^ { 2 } + 5 x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 12 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the quadratic factor of \(\mathrm { p } ( x )\).

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

1. Given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), find the value of \(a\).
2. When \(a\) has this value,
   1. factorise \(\mathrm { p } ( x )\) completely,
   2. find all the roots of the equation \(\mathrm { p } \left( x ^ { 2 } \right) = 0\).

5 The polynomial \(8 x ^ { 3 } + a x ^ { 2 } + b x + 3\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by ( \(2 x - 1\) ) the remainder is 1 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the remainder when \(\mathrm { p } ( x )\) is divided by \(2 x ^ { 2 } - 1\).

6 It is given that \(2 \ln ( 4 x - 5 ) + \ln ( x + 1 ) = 3 \ln 3\).

1. Show that \(16 x ^ { 3 } - 24 x ^ { 2 } - 15 x - 2 = 0\).
2. By first using the factor theorem, factorise \(16 x ^ { 3 } - 24 x ^ { 2 } - 15 x - 2\) completely.
3. Hence solve the equation \(2 \ln ( 4 x - 5 ) + \ln ( x + 1 ) = 3 \ln 3\).

1 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 3 - x ) ( 1 + 3 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).

6 \includegraphics[max width=\textwidth, alt={}, center]{87392b1c-3683-45b4-8d55-36760b5f0cc1-10_547_531_260_806} The diagram shows the curve \(y = x ^ { 4 } - 2 x ^ { 3 } - 7 x - 6\). The curve intersects the \(x\)-axis at the points \(( a , 0 )\) and \(( b , 0 )\), where \(a < b\). It is given that \(b\) is an integer.

1. Find the value of \(b\).
2. Hence show that \(a\) satisfies the equation \(a = - \frac { 1 } { 3 } \left( 2 + a ^ { 2 } + a ^ { 3 } \right)\).
3. Use an iterative formula based on the equation in part (ii) to determine \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

4 The polynomial \(4 x ^ { 3 } + a x + 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value,
   1. factorise \(\mathrm { p } ( x )\),
   2. solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.

3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 4\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x - 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\). When \(a\) has this value,
2. factorise \(\mathrm { p } ( x )\),
3. solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.

1 Find the exact value of the constant \(k\) for which \(\int _ { 1 } ^ { k } \frac { 1 } { 2 x - 1 } \mathrm {~d} x = 1\).

5 The polynomial \(4 x ^ { 3 } - 4 x ^ { 2 } + 3 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(2 x ^ { 2 } - 3 x + 3\).

1. Find the value of \(a\).
2. When \(a\) has this value, solve the inequality \(\mathrm { p } ( x ) < 0\), justifying your answer.

5 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x - 4\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). The result of differentiating \(\mathrm { p } ( x )\) with respect to \(x\) is denoted by \(\mathrm { p } ^ { \prime } ( x )\). It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and of \(\mathrm { p } ^ { \prime } ( x )\).

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

10 The polynomial \(\mathrm { p } ( z )\) is defined by $$\mathrm { p } ( z ) = z ^ { 3 } + m z ^ { 2 } + 24 z + 32$$ where \(m\) is a constant. It is given that \(( z + 2 )\) is a factor of \(\mathrm { p } ( z )\).

1. Find the value of \(m\).
2. Hence, showing all your working, find
   1. the three roots of the equation \(\mathrm { p } ( z ) = 0\),
   2. the six roots of the equation \(\mathrm { p } \left( z ^ { 2 } \right) = 0\).

3 The polynomial \(x ^ { 4 } + 3 x ^ { 3 } + a x + 3\) is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(x ^ { 2 } - x + 1\).

1. Find the value of \(a\).
2. When \(a\) has this value, find the real roots of the equation \(\mathrm { p } ( x ) = 0\).

7 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - x ^ { 2 } + 4 x - a$$ where \(a\) is a constant. It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\).
2. When \(a\) has the value found in part (i), express \(\frac { 8 x - 13 } { \mathrm { p } ( x ) }\) in partial fractions.

3 The polynomial \(\mathrm { f } ( x )\) is defined by $$f ( x ) = x ^ { 3 } + a x ^ { 2 } - a x + 14$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).

1. Find the value of \(a\).
2. Show that, when \(a\) has this value, the equation \(\mathrm { f } ( x ) = 0\) has only one real root.

3 The polynomial \(a x ^ { 3 } + b x ^ { 2 } + x + 3\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 3 x + 1 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) the remainder is 21 . Find the values of \(a\) and \(b\).

3 The polynomial \(4 x ^ { 3 } + a x ^ { 2 } + b x - 2\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) and \(( x + 2 )\) are factors of \(\mathrm { p } ( x )\).

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the remainder when \(\mathrm { p } ( x )\) is divided by \(\left( x ^ { 2 } + 1 \right)\).

6 The polynomial \(8 x ^ { 3 } + a x ^ { 2 } + b x - 1\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { p } ( x )\) and that when \(\mathrm { p } ( x )\) is divided by \(( 2 x + 1 )\) the remainder is 1 .

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

7

1. Show that ( \(x + 1\) ) is a factor of \(4 x ^ { 3 } - x ^ { 2 } - 11 x - 6\).
2. Find \(\int \frac { 4 x ^ { 2 } + 9 x - 1 } { 4 x ^ { 3 } - x ^ { 2 } - 11 x - 6 } \mathrm {~d} x\).