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

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 2

HLPaper 1

Consider the quadratic equation $(2k+1)x^2 - 4kx + (k-2) = 0$ , where $k$ is a real number.

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 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 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 polynomial $f(x) = 2x^3 - 3x^2 - 11x + 6$ .

Use the factor theorem to determine if $x - 1$ is a factor of $f(x)$ .

[3]

Find the remainder when $f(x)$ is divided by $x + 2$ .

[3]

Factorize $f(x)$ completely.

[4]

Question 6

HLPaper 2

Consider $f(x)=\frac{x^3+4x^2+x+\lambda}{x^2 - x - 6},\qquad x\in\mathbb{R}\setminus\{-2,3\},\ \lambda\in\mathbb{R}\setminus\{-6, -66\}.$

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

The function $f$ is defined by $f(x)=\frac{x^3+2x^2+6x+\lambda}{x^2-2x-3}$ for $x \in \mathbb{R}, x \neq -1, 3$ , where $\lambda \in \mathbb{R}$ and $\lambda \neq 5, -63$ .

Find the equations of the vertical asymptotes and justify why they are independent of $\lambda$ .

[2]

Express $f(x)$ in the form $f(x)=q(x)+\dfrac{r(x)}{x^2-2x-3}$ , where $q(x)$ is a linear function.

[4]

State the equation of the slant asymptote.

[2]

Find the coordinates of the point where the graph of $f$ intersects its slant asymptote.

[4]

Given that $\lambda=-12$ , solve the inequality $f(x) \ge x+4$ .

[6]

Question 8

HLPaper 1

Consider the polynomial $r(x) = 3x^3 - 11x^2 + mx + 8$ .

Given that $r(x)$ has a factor $x - 4$ , find the value of $m$ .

[3]

Hence or otherwise, factorize $r(x)$ as a product of linear factors.

[3]

Question 9

HLPaper 2

Consider the quadratic equation $mx^2 - (m + 3)x + 2m + 9 = 0$ , where $m \in \mathbb{R}$ .

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

[1]

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

[3]

Find for what values there is one distinct real root.

[3]

Question 10

HLPaper 2

Let

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

Identify the vertical asymptotes.

[2]

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

[4]

State the equation of the slant asymptote.

[2]

Find the coordinates of the point of intersection between the graph of $f$ and the slant asymptote, in terms of $\lambda$ .

[4]

For $\lambda = 3$ , solve the inequality $f(x) \ge x + 1$ .

[6]