Given that ${x^2} - 1$ is a factor of $f(x)$ find the value of $a$ and the value of $b$.

Factorize $f(x)$ into a product of linear factors.

Sketch the graph of $y = f(x)$, labelling the maximum and minimum points and the $x$ and $y$ intercepts.

Using your graph state the range of values of $c$ for which $f(x) = c$ has exactly two distinct real roots.

$g(x) = 3{x^4} + a{x^3} + b{x^2} - 7x - 4$

$g(1) = 0 \Rightarrow a + b = 8$     M1A1

$g( - 1) = 0 \Rightarrow - a + b = - 6$     A1

$\Rightarrow a = 7,{\text{ }}b = 1$     A1

[4 marks]

$3{x^4} + 7{x^3} + {x^2} - 7x - 4 = ({x^2} - 1)(p{x^2} + qx + r)$

attempt to equate coefficients     (M1)

$p = 3,{\text{ }}q = 7,{\text{ }}r = 4$     (A1)

$3{x^4} + 7{x^3} + {x^2} - 7x - 4 = ({x^2} - 1)(3{x^2} + 7x + 4)$

$= (x - 1){(x + 1)^2}(3x + 4)$     A1

Note:     Accept any equivalent valid method.

[3 marks]

A1 for correct shape (ie with correct number of max/min points)

A1 for correct $x$ and $y$ intercepts

A1 for correct maximum and minimum points

[3 marks]

$c > 0$     A1

$- 6.20 < c < - 0.0366$     A1A1

Note:     Award A1 for correct end points and A1 for correct inequalities.

Note:     If the candidate has misdrawn the graph and omitted the first minimum point, the maximum mark that may be awarded is A1FTA0A0 for $c > - 6.20$ seen.

[3 marks]