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]