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

5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } + b x ^ { 2 } - a x + 8$$ where \(a\) and \(b\) are constants.It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) ,and that the remainder is 24 when \(\mathrm { p } ( x )\) is divided by \(( x - 2 )\) .

1. Find the values of \(a\) and \(b\) . \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-09_2723_35_101_20}
2. Factorise \(\mathrm { p } ( x )\) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.
3. Solve the equation \(\mathrm { p } \left( \frac { 1 } { 2 } \operatorname { cosec } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-10_499_696_264_680} The diagram shows the curves with equations \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and \(y = \frac { 27 } { 2 x + 5 }\) for \(x \geqslant 0\).
   The curves meet at the point \(( 2,3 )\).
   Region \(A\) is bounded by the curve \(y = \sqrt [ 3 ] { 5 x ^ { 2 } + 7 }\) and the straight lines \(x = 0 , x = 2\) and \(y = 0\).
   Region \(B\) is bounded by the two curves and the straight line \(x = 0\).

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 \(( x + 3 )\).

2 The cubic polynomial \(3 x ^ { 3 } + a x ^ { 2 } - 2 x - 8\) is denoted by \(\mathrm { f } ( x )\).

1. Given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. When \(a\) has this value, factorise \(\mathrm { f } ( x )\) completely.

2 The polynomial \(x ^ { 4 } - 9 x ^ { 2 } - 6 x - 1\) is denoted by \(\mathrm { f } ( x )\).

1. Find the value of the constant \(a\) for which $$f ( x ) \equiv \left( x ^ { 2 } + a x + 1 \right) \left( x ^ { 2 } - a x - 1 \right)$$
2. Hence solve the equation \(\mathrm { f } ( x ) = 0\), giving your answers in an exact form.

3 The cubic polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 13 x - 6\) is denoted by \(\mathrm { f } ( x )\). It is given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value, solve the equation \(\mathrm { f } ( x ) = 0\).

4 The polynomial \(x ^ { 3 } - x ^ { 2 } + a x + b\) 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 \(( x - 2 )\) the remainder is 12 .

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

4 The cubic polynomial \(a x ^ { 3 } + b x ^ { 2 } - 3 x - 2\), 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 other linear factor of \(\mathrm { p } ( x )\).

4 The polynomial \(2 x ^ { 3 } - 3 x ^ { 2 } + a x + b\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) the remainder is - 20 .

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 ( \(x ^ { 2 } - 4\) ).

4 The polynomial \(2 x ^ { 3 } + 7 x ^ { 2 } + a x + b\), 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 \(( x + 2 )\) the remainder is 5 . Find the values of \(a\) and \(b\).

6 The polynomial \(x ^ { 3 } + a x ^ { 2 } + b x + 6\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x - 1 )\) the remainder is 4 .

1. Find the values of \(a\) and \(b\).
2. When \(a\) and \(b\) have these values, find the other two linear factors of \(\mathrm { p } ( x )\).

4 The polynomial \(x ^ { 3 } + 3 x ^ { 2 } + 4 x + 2\) is denoted by \(\mathrm { f } ( x )\).

1. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(x ^ { 2 } + x - 1\).
2. Use the factor theorem to show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).

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.

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

1. Given that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\), find the value of \(a\).
2. When \(a\) has the value found in part (i), find the quotient when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).

7 The cubic polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + b x + 10$$ where \(a\) and \(b\) are constants. It is given that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) and that, when \(\mathrm { p } ( x )\) is divided by ( \(x + 1\) ), the remainder is 24 .

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

3 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - 3 x ^ { 2 } - 5 x + a + 4 ,$$ 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 the remainder when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).

3

1. Find the quotient when the polynomial $$8 x ^ { 3 } - 4 x ^ { 2 } - 18 x + 13$$ is divided by \(4 x ^ { 2 } + 4 x - 3\), and show that the remainder is 4 .
2. Hence, or otherwise, factorise the polynomial $$8 x ^ { 3 } - 4 x ^ { 2 } - 18 x + 9$$

4 The polynomial \(a x ^ { 3 } - 5 x ^ { 2 } + b x + 9\), where \(a\) and \(b\) are constants, is denoted by \(\mathrm { p } ( x )\). It is given that \(( 2 x + 3 )\) is a factor of \(\mathrm { p } ( x )\), and that when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\) the remainder is 8 .

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

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 \(( x + 3 )\).

3

1. Find the quotient when \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 15\) is divided by ( \(x ^ { 2 } - 4\) ), and confirm that the remainder is 7 .
2. Hence solve the equation \(6 x ^ { 4 } - x ^ { 3 } - 26 x ^ { 2 } + 4 x + 8 = 0\).

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = x ^ { 3 } + 2 x + a$$ 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, find the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x + 2\) ) and hence show that the equation \(\mathrm { p } ( x ) = 0\) has exactly one real root.

4 The polynomials \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined by $$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 3 } + b x ^ { 2 } - a$$ where \(a\) and \(b\) are constants. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\). It is also given that, when \(\mathrm { g } ( x )\) 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, find the greatest possible value of \(\mathrm { g } ( x ) - \mathrm { f } ( x )\) as \(x\) varies.

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 8 x ^ { 3 } + 30 x ^ { 2 } + 13 x - 25$$

1. Find the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x + 2\) ), and show that the remainder is 5 .
2. Hence factorise \(\mathrm { p } ( x ) - 5\) completely.
3. Write down the roots of the equation \(\mathrm { p } ( | x | ) - 5 = 0\).

2

1. Find the quotient and remainder when \(2 x ^ { 3 } - 7 x ^ { 2 } - 9 x + 3\) is divided by \(x ^ { 2 } - 2 x + 5\).
2. Hence find the values of the constants \(p\) and \(q\) such that \(x ^ { 2 } - 2 x + 5\) is a factor of \(2 x ^ { 3 } - 7 x ^ { 2 } + p x + q\).

6

1. Use the factor theorem to show that ( \(x + 2\) ) is a factor of the expression $$6 x ^ { 3 } + 13 x ^ { 2 } - 33 x - 70$$ and hence factorise the expression completely.
2. Deduce the roots of the equation $$6 + 13 y - 33 y ^ { 2 } - 70 y ^ { 3 } = 0$$

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by $$\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + 14 x + a + 1$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { f } ( x )\).

1. Use the factor theorem to find the value of \(a\) and hence factorise \(\mathrm { f } ( x )\) completely.
2. Hence, without using a calculator, solve the equation \(\mathrm { f } ( 2 x ) = 3 \mathrm { f } ( x )\).