Polynomials & Quadratics — Higher Maths Practice Questions

1

(a) Show that

\displaystyle 2x+3

is a factor of

\displaystyle 2x^3+3x^2-2x-3.

(b) Hence factorise

\displaystyle 2x^3+3x^2-2x-3

fully.

No hints on this one — its full written solution is with the answer.

2

Factorise

\displaystyle 2x^3+9x^2-6x-5

fully.

No hints on this one — its full written solution is with the answer.

3

(a) Show that

\displaystyle x+2

is a factor of

\displaystyle 2x^4+9x^3-x^2-18x+8.

(b) Hence factorise

\displaystyle 2x^4+9x^3-x^2-18x+8

fully.

No hints on this one — its full written solution is with the answer.

4

(a) Show that

\displaystyle x-3

is a factor of

\displaystyle 2x^4-3x^3-19x-24.

(b) Hence factorise

\displaystyle 2x^4-3x^3-19x-24

fully.

No hints on this one — its full written solution is with the answer.

5

For the polynomial

\displaystyle x^3-x^2+mx+n

:
•

\displaystyle x-3

is a factor
• 54 is the remainder when it is divided by

\displaystyle x-5

(a) Determine the values of

\displaystyle m

and

\displaystyle n.

(b) Hence solve

\displaystyle x^3-x^2+mx+n=0.

No hints on this one — its full written solution is with the answer.

6

The same remainder is found when

\displaystyle x^3-6x^2+2x-p

and

\displaystyle x^3+5x^2+(2p+1)x-37

are divided by

\displaystyle (x+2).

Find the value of

\displaystyle p.

No hints on this one — its full written solution is with the answer.

7

Solve

\displaystyle x^3+3x^2-4x-12=0.

No hints on this one — its full written solution is with the answer.

8

Solve

\displaystyle x^4-x^3-10x^2+4x+24=0.

No hints on this one — its full written solution is with the answer.

9

The graph of

\displaystyle y=f(x),

where

\displaystyle f(x)=k(x-a)(x-b)^{2},

has a minimum turning point at

\displaystyle (3,0)

, a root

\displaystyle -2

and passes through the point

\displaystyle (1,48).

Find the values of

\displaystyle a

,

\displaystyle b

and

\displaystyle k.

No hints on this one — its full written solution is with the answer.

10

Find the values of

\displaystyle k

for which

\displaystyle x^2+(k+3)x+4=0

has equal roots.

No hints on this one — its full written solution is with the answer.

11

Find the range of values of

\displaystyle p

for which

\displaystyle 2x^2+5x+p+1=0

has no real roots.

No hints on this one — its full written solution is with the answer.

12

Find the range of values of

\displaystyle a

for which

\displaystyle x^2-6x+a=0

has two distinct real roots.

No hints on this one — its full written solution is with the answer.

13

Find the range of values of

\displaystyle n

for which

\displaystyle x^2-nx+3-n=0

has two distinct real roots.

No hints on this one — its full written solution is with the answer.

14

A rectangle has length

\displaystyle x

cm and a breadth that is

\displaystyle 1

cm shorter than the length. Its area is less than

\displaystyle 30

cm

\displaystyle ^2.

Find the range of possible values of

\displaystyle x.

No hints on this one — its full written solution is with the answer.

15

Express

\displaystyle -2x^2+12x+5

in the form

\displaystyle a(x+b)^2+c.

No hints on this one — its full written solution is with the answer.

16

Express

\displaystyle 4x^2-28x-1

in the form

\displaystyle p(x+q)^2+r.

No hints on this one — its full written solution is with the answer.

17

Determine the point(s) of intersection of the parabola

\displaystyle y=x^2+3x-7

and the line

\displaystyle y=4x-1.

No hints on this one — its full written solution is with the answer.

18

The line

\displaystyle y=5x-3

and the curve

\displaystyle y=x^3-8x+9

intersect at three points. One of these points is

\displaystyle (3, 12).

Find the coordinates of the other two points of intersection.

No hints on this one — its full written solution is with the answer.

19

2023 P1 Q5

3 Marks

The equation $\displaystyle 2x^{2}+(3p-2)x+p=0$ has equal roots.
Determine the possible values of p.

20

2023 P1 Q10

(2, 5)7 Marks

(a) Show that $\displaystyle (x+5)$ is a factor of
$\displaystyle x^{4}+3x^{3}-7x^{2}+9x-30$
(b) Hence, or otherwise, solve
$\displaystyle x^{4}+3x^{3}-7x^{2}+9x-30=0$
$\displaystyle x\in\mathbb{R}.$

21

2024 P1 Q8

4 Marks

The equation $\displaystyle x^{2}+(m-4)x+(2m-3)=0$ has no real roots.
Determine the range of values for m.
Justify your answer.

22

2024 P1 Q10

(2, 4)6 Marks

(a) Show that $\displaystyle (x-1)$ is a factor of
$\displaystyle 2x^{4}+3x^{3}-4x^{2}-3x+2$
(b) Hence, or otherwise, factorise $\displaystyle 2x^{4}+3x^{3}-4x^{2}-3x+2$ fully.

23

2025 P1 Q7

(2, 3)5 Marks

(a) Show that $\displaystyle (x+3)$ is a factor of $\displaystyle 5x^{3}+16x^{2}-x-12$
(b) Hence, or otherwise, solve
$\displaystyle 5x^{3}+16x^{2}-x-12=0.$

24

2025 P1 Q11

4 Marks

The equation $\displaystyle 9x^{2}+3kx+k=0$ has two real and distinct roots.
Determine the range of values for k.
Justify your answer.

25

2025 P2 Q2

3 Marks

Express

\displaystyle 2x^{2}+16x+5

in the form

\displaystyle p(x+q)^{2}+r

26

2026 P1 Q11

(2, 2, 4)8 Marks

(a)
    (i)  Show that $\displaystyle (x+2)$ is a factor of $\displaystyle x^{3}+7x^{2}+18x+16.$
    (ii)  Explain why $\displaystyle (x+2)$ is the only linear factor of $\displaystyle x^{3}+7x^{2}+18x+16.$
    Give a reason for your answer.
(b)  Find the coordinates of the point(s) where the curves with equations
$\displaystyle y=2x^{3}+20x^{2}+27x+9$ and $\displaystyle y=6x^{2}-9x-23$
intersect.

27

2026 P2 Q1

3 Marks

The equation

\displaystyle x^{2}+kx+(k+3)=0

has equal roots.
Find algebraically the values of k.