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Polynomials & Quadratics · Guided Practice

Polynomials & Quadratics

27 questions with answers and video solutions. Try each one before revealing the answer.

1
(a)  Show that 2x+3\displaystyle 2x+3 is a factor of 2x3+3x2−2x−3.\displaystyle 2x^3+3x^2-2x-3.
(b)  Hence factorise 2x3+3x2−2x−3\displaystyle 2x^3+3x^2-2x-3 fully.
2
Factorise 2x3+9x2−6x−5\displaystyle 2x^3+9x^2-6x-5 fully.
3
(a)  Show that x+2\displaystyle x+2 is a factor of 2x4+9x3−x2−18x+8.\displaystyle 2x^4+9x^3-x^2-18x+8.
(b)  Hence factorise 2x4+9x3−x2−18x+8\displaystyle 2x^4+9x^3-x^2-18x+8 fully.
4
(a)  Show that x−3\displaystyle x-3 is a factor of 2x4−3x3−19x−24.\displaystyle 2x^4-3x^3-19x-24.
(b)  Hence factorise 2x4−3x3−19x−24\displaystyle 2x^4-3x^3-19x-24 fully.
5
For the polynomial x3−x2+mx+n\displaystyle x^3-x^2+mx+n:
• x−3\displaystyle x-3 is a factor
• 54 is the remainder when it is divided by x−5\displaystyle x-5
(a)  Determine the values of m\displaystyle m and n.\displaystyle n.
(b)  Hence solve x3−x2+mx+n=0.\displaystyle x^3-x^2+mx+n=0.
6
The same remainder is found when x3−6x2+2x−p\displaystyle x^3-6x^2+2x-p and x3+5x2+(2p+1)x−37\displaystyle x^3+5x^2+(2p+1)x-37 are divided by (x+2).\displaystyle (x+2).
Find the value of p.\displaystyle p.
7
Solve x3+3x2−4x−12=0.\displaystyle x^3+3x^2-4x-12=0.
8
Solve x4−x3−10x2+4x+24=0.\displaystyle x^4-x^3-10x^2+4x+24=0.
9
The graph of y=f(x),\displaystyle y=f(x), where f(x)=k(x−a)(x−b)2,\displaystyle f(x)=k(x-a)(x-b)^{2}, has a minimum turning point at (3,0)\displaystyle (3,0), a root −2\displaystyle -2 and passes through the point (1,48).\displaystyle (1,48).
Find the values of a\displaystyle a, b\displaystyle b and k.\displaystyle k.
10
Find the values of k\displaystyle k for which x2+(k+3)x+4=0\displaystyle x^2+(k+3)x+4=0 has equal roots.
11
Find the range of values of p\displaystyle p for which 2x2+5x+p+1=0\displaystyle 2x^2+5x+p+1=0 has no real roots.
12
Find the range of values of a\displaystyle a for which x2−6x+a=0\displaystyle x^2-6x+a=0 has two distinct real roots.
13
Find the range of values of n\displaystyle n for which x2−nx+3−n=0\displaystyle x^2-nx+3-n=0 has two distinct real roots.
14
A rectangle has length x\displaystyle x cm and a breadth that is 1\displaystyle 1 cm shorter than the length. Its area is less than 30\displaystyle 30 cm2.\displaystyle ^2.
Find the range of possible values of x.\displaystyle x.
15
Express −2x2+12x+5\displaystyle -2x^2+12x+5 in the form a(x+b)2+c.\displaystyle a(x+b)^2+c.
16
Express 4x2−28x−1\displaystyle 4x^2-28x-1 in the form p(x+q)2+r.\displaystyle p(x+q)^2+r.
17
Determine the point(s) of intersection of the parabola y=x2+3x−7\displaystyle y=x^2+3x-7 and the line y=4x−1.\displaystyle y=4x-1.
18
The line y=5x−3\displaystyle y=5x-3 and the curve y=x3−8x+9\displaystyle y=x^3-8x+9 intersect at three points. One of these points is (3,12).\displaystyle (3, 12).
Find the coordinates of the other two points of intersection.
19
2023 P1 Q5
3 Marks

The equation 2x2+(3p−2)x+p=0\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 (x+5)\displaystyle (x+5) is a factor of
x4+3x3−7x2+9x−30\displaystyle x^{4}+3x^{3}-7x^{2}+9x-30
(b) Hence, or otherwise, solve
x4+3x3−7x2+9x−30=0\displaystyle x^{4}+3x^{3}-7x^{2}+9x-30=0
x∈R.\displaystyle x\in\mathbb{R}.

21
2024 P1 Q8
4 Marks

The equation x2+(m−4)x+(2m−3)=0\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 (x−1)\displaystyle (x-1) is a factor of
2x4+3x3−4x2−3x+2\displaystyle 2x^{4}+3x^{3}-4x^{2}-3x+2
(b)  Hence, or otherwise, factorise 2x4+3x3−4x2−3x+2\displaystyle 2x^{4}+3x^{3}-4x^{2}-3x+2 fully.

23
2025 P1 Q7
(2, 3)5 Marks

(a) Show that (x+3)\displaystyle (x+3) is a factor of 5x3+16x2−x−12\displaystyle 5x^{3}+16x^{2}-x-12
(b) Hence, or otherwise, solve
5x3+16x2−x−12=0.\displaystyle 5x^{3}+16x^{2}-x-12=0.

24
2025 P1 Q11
4 Marks

The equation 9x2+3kx+k=0\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 2x2+16x+5\displaystyle 2x^{2}+16x+5
in the form p(x+q)2+r\displaystyle p(x+q)^{2}+r
26
2026 P1 Q11
(2, 2, 4)8 Marks

(a)
    (i)  Show that (x+2)\displaystyle (x+2) is a factor of x3+7x2+18x+16.\displaystyle x^{3}+7x^{2}+18x+16.
    (ii)  Explain why (x+2)\displaystyle (x+2) is the only linear factor of x3+7x2+18x+16.\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
y=2x3+20x2+27x+9\displaystyle y=2x^{3}+20x^{2}+27x+9 and y=6x2−9x−23\displaystyle y=6x^{2}-9x-23
intersect.

27
2026 P2 Q1
3 Marks
The equation x2+kx+(k+3)=0\displaystyle x^{2}+kx+(k+3)=0 has equal roots.
Find algebraically the values of k.

Practice questions courtesy of Maths.scot. Full written solutions are on his site — the links above go straight to them.

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