Exercises for Question Type 2: For small polynomials, given a remainder or factor, finding a parameter's value

Question 1

Skill question

Determine $r$ such that $x^{4} + r x^{2} - x + 4$ has remainder $0$ when divided by $x - 1$ .

[3]

Question 2

Skill question

The polynomial $4 x^{3} + a x^{2} - 7 x + 1$ has a factor $x - 1$ . Find the value of $a$ .

[3]

Question 3

Skill question

Given $g (x) = 6 x^{4} + d x^{3} - x + 5$ leaves a remainder of $7$ when divided by $(x - 1)$ , find the value of $d$ .

[3]

Question 4

Skill question

The polynomial $Q (x) = x^{4} - 4 x^{2} + k x + 5$ gives a remainder of $13$ when divided by $(x - 2)$ . Find the value of $k$ .

[3]

Question 5

Skill question

Determine the value of $b$ given that $P (x) = 2 x^{4} - 3 x^{2} + b x - 6$ leaves a remainder of $5$ when divided by $(x - 2)$ .

[3]

Question 6

Skill question

Given that $P (x) = x^{3} + a x^{2} - x + 2$ leaves a remainder of 5 when divided by $(x - 1)$ , find the value of $a$ .

[3]

Question 7

Skill question

Find $c$ if $f (x) = x^{3} + c x^{2} - 4 x + 3$ is exactly divisible by $x - 3$ . [3 marks]

[3]

Question 8

Skill question

If $5 x^{3} - m x^{2} + 2 x + 1$ gives a remainder of $- 3$ when $x = - 1$ , find the value of $m$ .

[4]

Question 9

Skill question

Find $k$ if the polynomial $2 x^{3} + k x^{2} - 5 x + 1$ is exactly divisible by $x - 1$ .

[3]

Question 10

Skill question

Find $k$ such that $x^{5} + k x^{3} - x + 2$ is divisible by $x + 1$ .

[3]

Question 11

Skill question

Find the value of $b$ if the polynomial $3 x^{2} + b x + 10$ leaves a remainder of $3$ when divided by $(x - 2)$ .

[3]

Question 12

Skill question

The polynomial $h (x) = 3 x^{3} - p x + 4$ has a factor $x + 2$ . Determine the value of $p$ .

[3]

IB Mathematics Analysis and Approaches Question Type 2: for Small Polynomials, Given a Remainder or Factor, Finding a Parameter's Value Exercises