Polynomials, roots of polynomials & modulus inequalities – IB Maths exam questions with mark scheme

1CoreNon-calculatorHL only4 marks

(a) Find the remainder when $x^3-2x^2+x-5$ is divided by $(x-2)$. [2]

(b) Given that $(x-3)$ is a factor of $x^3-4x^2+px+6$, find the value of $p$. [2]

2StandardNon-calculatorHL only6 marks

Let $P(x)=2x^3-3x^2-11x+6$.

(a) Show that $(x-3)$ is a factor of $P(x)$. [2]

(b) Factorise $P(x)$ completely. [3]

$P(3)=54-27-33+6$M1
$=0$, so $(x-3)$ is a factorA1
Divides or compares to obtain $(x-3)(2x^2+bx+c)$M1
$(x-3)(2x^2+3x-2)$A1
$(x-3)(2x-1)(x+2)$A1
$x=3,\ \tfrac12,\ -2$A1
Answer: (b) $(x-3)(2x-1)(x+2)$ (c) $x=3,\tfrac12,-2$
Examiner tip: 'Show that' requires the substitution to be written out and the value 0 stated.

3StandardNon-calculatorHL only5 marks

The polynomial $P(x)=x^3+ax^2+bx-6$ has remainder 3 when divided by $(x-1)$ and has $(x+2)$ as a factor. Find the values of $a$ and $b$.

$P(1)=3$: $1+a+b-6=3$M1
$a+b=8$A1
$P(-2)=0$: $-8+4a-2b-6=0$M1
$2a-b=7$A1
$a=5,\ b=3$A1
Answer: $a=5,\ b=3$
Examiner tip: Remainder theorem gives an equation equal to the remainder; factor theorem gives an equation equal to zero.

4HardNon-calculatorHL only5 marks

The roots of $2x^3-7x^2+4x+3=0$ are $\alpha,\beta,\gamma$.

(a) Write down the values of $\alpha+\beta+\gamma$ and $\alpha\beta+\beta\gamma+\gamma\alpha$. [2]

(b) Find the value of $\alpha^2+\beta^2+\gamma^2$. [3]

$\alpha+\beta+\gamma=\tfrac72$A1
$\alpha\beta+\beta\gamma+\gamma\alpha=2$A1
$\alpha^2+\beta^2+\gamma^2=(\alpha+\beta+\gamma)^2-2(\alpha\beta+\beta\gamma+\gamma\alpha)$M1
$\tfrac{49}4-4$A1
$\tfrac{33}4$A1
Answer: (a) $\tfrac72$, $2$ (b) $\tfrac{33}4$
Examiner tip: The sum of roots is $-b/a$: the sign matters.

5HardNon-calculatorHL only5 marks

The roots of $2x^2-5x+1=0$ are $\alpha$ and $\beta$. Find a quadratic equation with integer coefficients whose roots are $\alpha^2$ and $\beta^2$.

$\alpha+\beta=\tfrac52$A1
$\alpha\beta=\tfrac12$A1
$\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$M1
Sum $=\tfrac{21}4$ and product $\alpha^2\beta^2=\tfrac14$A1
$4x^2-21x+1=0$A1
Answer: $4x^2-21x+1=0$
Examiner tip: Form $x^2-(\text{sum})x+\text{product}=0$, then clear fractions.

6HardNon-calculatorHL only6 marks

The equation $z^3-5z^2+az+b=0$, where $a,b\in\mathbb{R}$, has $1+2i$ as one root.

(a) Write down another non-real root of the equation. [1]

(b) Find the third root. [2]

$1-2i$ (non-real roots of a polynomial equation with real coefficients occur in conjugate pairs)A1
Sum of roots $=5$: $(1+2i)+(1-2i)+r=5$ (or divides the cubic by $z^2-2z+5$, or any valid method)M1
$r=3$A1
Valid method to find $a$ and $b$, e.g. expands $(z^2-2z+5)(z-r)$; or uses sum of pairwise products / product of roots; or substitutes $z=1+2i$ and equates real and imaginary parts ($4+a+b=0$, $2a-22=0$)M1
$a=11$ (FT their $r$ from (b))A1
$b=-15$ (FT their $r$ from (b))A1
Answer: (a) $1-2i$ (b) $3$ (c) $a=11$, $b=-15$
Examiner tip: For a cubic $z^3+pz^2+qz+c$, the sum of roots is $-p$ (here $5$, not $-5$); real coefficients force the conjugate root $1-2i$.

7HardNon-calculatorHL only5 marks

Solve the inequality $|2x-3|<|x+1|$.

Squares both sides: $(2x-3)^2<(x+1)^2$M1
$3x^2-14x+8<0$A1
Factorises $(3x-2)(x-4)<0$M1
Critical values $x=\tfrac23$ and $x=4$A1
$\tfrac23A1
Answer: $\tfrac23
Examiner tip: Squaring is valid because both sides are non-negative; check a test value inside and outside the interval.

8StretchNon-calculatorHL only8 marks

The diagram shows the graph of $y=|x+2|+|x-1|$ and the line $y=5$.

(a) Find the x-coordinates of the points where the graph meets the line $y=5$. [3]

(b) Hence solve $|x+2|+|x-1|<5$. [1]

Book a free introductory session and we'll find exactly which step is costing you.