17M.2.AHL.TZ2.H_11

pestleMathematicsAAHLPaper 217M· ahl-2-12-factor-and-remainder-theorems-sum-and-product-of-rootssource ↗

It is given that $f(x) = 3{x^4} + a{x^3} + b{x^2} - 7x - 4$ where $a$ and $b$ are positive integers.

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

[4]

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

[3]

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

[3]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

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

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

$c > 0$ 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]

Examiners’ report

[N/A]