AHL 2.12—Factor and remainder theorems, sum and product of roots Questionbank

Practice AHL 2.12—Factor and remainder theorems, sum and product of roots with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

The polynomial $p(x)=2 x^{3}-5 x^{2}+k x+6$ has a factor $(x-3)$ .

Determine the value of $k$ .

[3]

Hence, factorize $p(x)$ into a product of linear factors.

[3]

Question 2

HLPaper 1

Consider the quadratic equation $(2 k+1) x^{2}-4 k x+(k-2)=0$ , where $k \in \mathbb{R}$ .

Write down an expression for the product of the roots, in terms of $k$ .

[1]

Hence, or otherwise, determine the values of $k$ such that the equation has one positive and one negative real root, given that the roots are real.

[3]

Question 3

HLPaper 1

Consider the quadratic equation $k x^{2}+(k-4) x+k+2=0$ , where $k \in \mathbb{R}$ .

Write down an expression for the product of the roots, in terms of $k$ .

[1]

Hence, or otherwise, determine the values of $k$ such that the equation has one positive and one negative real root.

[3]

Question 4

HLPaper 1

The quadratic equation $x^{2}+k x+(k+2)=0$ has roots $\alpha$ and $\beta$ such that $\alpha^{2}+\beta^{2}=8$ .

Without solving the equation, find the possible values of the real number $k$ .

[6]

Question 5

HLPaper 1

Consider the quartic equation $w^4 + 4w^3 + 8w^2 + 80w + 400 = 0$ , $w \in \mathbb{C}$ . Two of the roots of this equation are $c + di$ and $d + ci$ , where $c, d \in \mathbb{Z}$ . Find the possible values of $c$ .

[8]

Question 6

HLPaper 2

Consider $f(x)=\frac{x^3+4x^2+x+\lambda}{x^2 - x - 6},\qquad x\in\mathbb{R}\setminus\{-2,3\},\ \lambda\in\mathbb{R}.$

Show the vertical asymptotes are independent of $\lambda$ and find them.

[2]

Write $f(x)=q(x)+\dfrac{r(x)}{x^2-x-6}$ with $q$ linear and $\deg r\le1$ .

[4]

Deduce the oblique asymptote.

[2]

Find the intersection point(s) of the graph with its oblique asymptote.

[4]

Solve $f(x)\ge x+5$ for $\lambda=-30$ .

[6]

Question 7

HLPaper 2

Define $f(x)=\frac{x^3+2x^2+6x+\lambda}{x^2-2x-3},\qquad x\neq-1,3.$

Find the vertical asymptotes and justify independence from $\lambda$ .

[2]

Express $f(x)=q(x)+\dfrac{r(x)}{x^2-2x-3}$ with $q$ linear.

[4]

State the slant asymptote.

[2]

Find the intersection with $y=x+4$ .

[4]

Solve $f(x)\ge x+4$ for $\lambda=-12$ .

[6]

Question 8

HLPaper 2

Let $f(x)=\frac{x^3+5x^2 - x + \lambda}{x^2 + 4x + 3},\qquad x \neq -3,-1.$

Identify the vertical asymptotes.

[2]

Perform the division to obtain $f(x)=q(x)+\dfrac{r(x)}{x^2+4x+3}$ .

[4]

State the slant asymptote.

[2]

Find the coordinates of intersection with the slant asymptote.

[4]

Solve $f(x)\ge x+1$ for $\lambda=3$ .

[6]

Question 9

HLPaper 1

Let $P(x)=2 x^{4}-k x^{3}+5 x^{2}+3 x-2$ , where $k \in \mathbb{R}$ , be a polynomial with roots $\alpha, \beta, \gamma, \delta$ . The polynomial is divisible by $(x-1)^{2}$ .

Find the value of $k$ .

[4]

Given that the sum of the roots of the quotient polynomial after dividing $P(x)$ by $(x-1)^{2}$ is 3 , find the product of the roots of the quotient polynomial.

[4]

Question 10

HLPaper 1

The quadratic equation $2 x^{2}-4 k x+\left(k^{2}-1\right)=0$ has roots $\alpha$ and $\beta$ such that $\alpha^{2}-\beta^{2}=5$ .

Without solving the equation, find the possible values of the real number $k$ .

[7]