AHL 2.12—Factor and remainder theorems, sum and product of roots - IB Questionbank | RevisionDojo

SL 2.1—Equations of straight lines, parallel and perpendicular

SL 2.2—Functions, notation domain, range and inverse as reflection

SL 2.3—Graphing

SL 2.4—Key features of graphs, intersections using technology

SL 2.5—Composite functions, identity, finding inverse

SL 2.6—Quadratic function

SL 2.7—Solutions of quadratic equations and inequalities, discriminant and nature of roots

SL 2.8—Reciprocal and simple rational functions, equations of asymptotes

SL 2.9—Exponential and logarithmic functions

SL 2.10—Solving equations graphically and analytically

SL 2.11—Transformation of functions

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

AHL 2.13—Rational functions

AHL 2.14—Odd and even functions, self-inverse, inverse and domain restriction

AHL 2.15—Solutions of inequalities

AHL 2.16—Graphing modulus equations and inequalities

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 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 $f(x)$ .

[2]

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

[2]

Factorize $f(x)$ completely.

[4]

Question 9

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 10

HLPaper 2

Consider the equation $mx^2 - (m + 3)x + 2m + 9 = 0$ , where $m ∈ ℝ$ .

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]

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

[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 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 $f(x)$ .

[2]

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

[2]

Factorize $f(x)$ completely.

[4]

Question 9

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 10

HLPaper 2

Consider the equation $mx^2 - (m + 3)x + 2m + 9 = 0$ , where $m ∈ ℝ$ .

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]