Math topic中文

Remainder factor theorems

use of remainder and factor theorems, including factorising polynomials and solving cubic equations

Practice questions

1The expression $f(x) = ax^3 + (a - 3b)x^2 + 3bx + c$ is exactly divisible by $x^2 + 3x$ . When $f(x)$ is divided by $x + 2$ …
2The equation of a curve is $y=16-ax-x^{2}$ , where $a$ is positive constant. Given that $y$ can be expressed in the form…
3Show that 1 is a solution of the equation $x^3 - 5x^2 = 2 - 6x$ . Hence solve the equation $x^3 - 5x^2 = 2 - 6x$ complete…
4It is given that $f(x)=x^3+ax^2+bx+5$ , where $a$ and $b$ are constants, has a factor of $x-1$ and leaves a remainder of…
5It is given that $\frac{2^x \times 32(2^x)}{8^{x+1}} = \frac{9(5^{2x}) - 5^{2x+1}}{5^x - 5^{x-1}}$ . Evaluate $10^x$ with…
6A polynomial $\mathrm{f}(x)$ is given by $4x^3 - 6x^2 + kx + 3$ , where $k$ is a constant. Find the value of $k$ given th…
7Given that $f(x) = 2x^3 + px^2 + 7x + q$ . Find the value of $p$ and of $q$ for which $(x-3)$ and $(x-2)$ are factors of…
8When the height of liquid in a container is $h$ m, the volume of liquid is $V\text{ m}^3$ where $V = 0.06 \left[ (4h + 1)^3 - 7 \right] \text{ and } h \ge 0.25$ …
9Express $\dfrac{2x^{2}+3x}{x^{2}+3x+2}$ in partial fractions. Hence evaluate $\displaystyle \int_{1}^{3} \dfrac{2x^{2}+3x}{x^{2}+3x+2}\,dx$ …
10The function $f$ is defined by $f(x) = 3x^3 - 5cx^2 + kc^2x + 4c^3$ , where $c$ and $k$ are non-zero constants. $f(x)$ le…
11Find, in ascending powers of $x$ , the first three terms in the expansion of $(1+4x)^7$ , $(2-x)^7$ . Hence, find the coeff…
12The polynomial $f(x)$ is given by $f(x)=9x^3-30x^2-23x-4$ . Factorise $f(x)$ completely. Hence, prove that the equation $9x\sqrt{x}-23\sqrt{x}=4+30x$ …
13The equation of a curve is $y = e^{-2x}\tan x$ . Show that $\dfrac{dy}{dx} = e^{-2x}(1 - \tan x)^2$ . Find, in terms of $\pi$ …
14The remainder when $x^3 + 17x^2 + 7x - 9$ is divided by $x - a$ is twice the remainder when it is divided by $x + a$ . Sh…
15Given that $y = 3e^{2x} + 2e^{-x}$ , and that $\frac{\text{d}^2 y}{\text{d}x^2} + \frac{\text{d}y}{\text{d}x} + y = Ae^{2x} + Be^{-x}$ …
16It is given that $f(x) = ax^3 + 3x^2 + bx + 1$ where $a$ and $b$ are constants. Given that $f(x)$ is divisible by $x + 1$ …
17Factorise $64 - (x + 1)^3$ . Hence, solve $64 - (x + 1)^3 = 15(3 - x)$ , expressing non-integer roots in surd form.
18The cubic polynomial $\text{f}(x)$ is such that the coefficient of $x^3$ is $-1$ and the roots of $\text{f}(x) = 0$ are…
19Show that $x+2$ is a solution of the equation $2x^3 + 5x^2 + x - 2 = 0$ and hence solve the equation completely.
20Factorise $x^3 - 125$ . The expression $-2x^3 + 7x^2 + ax + b$ , where $a$ and $b$ are integers, has a factor of $x - 5$ a…
21The polynomial $\mathrm{f}(x) = Ax^3 - (2A + B)x^2 + 2x + B$ , where $A$ and $B$ are constants, is exactly divisible by $(x - 1)$ …
22When $f(x)$ is divided by $x - 1$ , the remainder is $61$ . The remainder is $-123$ when $f(x)$ is divided by $x + 3$ . Fin…
23The polynomial $f(x)$ is such that $f(x)=6x^3+ax^2-50x+b$ , where $a$ and $b$ are integers. It is given that $f(x)$ is di…
24The equation of a polynomial is given by $\mathrm{p}(x) = 4x^3 + x - 5$ . Find the remainder when $\mathrm{p}(x)$ is divi…
25A particle P starts from rest at a fixed point O and moves in a straight line. The velocity, $v\ \text{ms}^{-1}$ , of the…
26Given that $\frac{(\log_y x)^2}{\log_x y} + 64 = 0$ , express $y$ in terms of $x$ . Solve the equation $\log_2 x + 3 = 2\log_2(x - 1)$ …
27The diagram shows the graph of $y = x^{3} - 3x^{2} - x - 12$ . Explain why $x^{3} - 3x^{2} - x - 12$ cannot be written as…
28When the expression $x^{3}+2kx+2$ is divided by $x+2$ , the remainder is 3 less than the remainder obtained when the expr…
29Show that $x - 2$ is a factor of $x^3 - 12x + 16$ . Express $\frac{2 + x^2}{x^3 - 12x + 16}$ as the sum of three partia…
30The diagram shows a cat scratch stand which consists of a solid cylinder fixed to a solid cuboid. The cylinder has a rad…
31It is given that $f(x)=2x^3+ax^2+bx-10$ , where $a$ and $b$ are constants, has a factor of $(x-1)^2$ . Find the value of $a$ …
32Show that $x - 2$ is a factor of $x^{3} - 2x^{2} + 3x - 6$ . Express $\dfrac{11 - 2x}{x^{3} - 2x^{2} + 3x - 6}$ as partia…
33Given that $\text{p}(x) = (x+1)^3 + 5(x-1)^2 + ax + b$ has a factor of $x-1$ and leaves a remainder of 21 when divided b…
34Given that $\text{p}(x) = (x + 1)^3 + 5(x - 1)^2 + ax + b$ has a factor $x - 1$ and it leaves a remainder of 21 when div…
35Complete the table for $y = 9x^3 + 5x^2 - 3x + 1$ . Give your answer correct to 1 decimal place. | $x$ | $-1$ | $-0.75$ |…
36By using long division, show that $2x^2+1$ is a factor of $2x^3-4x^2+x-2$ . Express $\dfrac{-5x^2+11x-11}{2x^3-4x^2+x-2}$ …
37Given that $x^2 + 2x - 3$ is a factor of $\text{f}(x)$ , where $\text{f}(x) = x^4 + 4x^3 + 2x^2 + ax - b$ . Find the value…
38Solve $2x^3 - x^2 - 15x - 10 = 0$ , showing all necessary working. Solve the equation $x - 3x + 1 = 0$ .
39Find the remainder when $3x^3 - x^2 + 4x + 2$ is divided by $x^2 + 1$ . The remainder when $x^3 + ax$ , where $a$ is a con…
40The polynomial $P(x) = 5x^3 + ax^2 - x + b$ , where $a$ and $b$ are constants is exactly divisible by $x^2 + 4x + 3$ . Sho…
41Find the remainder when $6x^3 - 13x^2 + 17x - 6$ is divided by $2x - 1$ . Show that there is only one real root of the eq…
42Factorise $x^4 - 1$ completely. Given that $\frac{x^4}{x^4 - 1} = a + \frac{b}{x^4 - 1}$ , find the values of $a$ and of…
43Express $\frac{9x^2 + 17x - 16}{3x^2 + 5x - 2}$ in partial fractions. Hence, find $\int \frac{9x^2 + 17x - 16}{3x^2 + 5x - 2} \, dx$ …
44The expression $2x^3 + ax^2 + bx + 10$ , where $a$ and $b$ are constants, has a factor of $x + 2$ and leaves a remainder…
45The expression $x^3-4x^2+ax+b$ , where $a$ and $b$ are constants, has a factor of $x+1$ and leaves a remainder of $-60$ w…
46Given that $\text{f}(x) = x^3 + 3x^2 - 4$ , show that $x+2$ is a factor of $\text{f}(x)$ , and hence factorise $\text{f}(x)$ …
47Factorise completely $2x^3+7x^2+4x-4$ . Express $\dfrac{12x^2+32x+31}{2x^3+7x^2+4x-4}$ in partial fractions.
48 $f(x)$ is such that $f''(x) = 18x - 4$ . Given that $f(0) = 1$ and $f(2) = 9$ , find $f(x)$ . Show that $x + 1$ is a factor…
49Integrate $\left(\frac{6}{e^{3-x}}\right)^2$ with respect to $x$ . Explain, with working, why the equation $2x^3 + 3x^2 + 2x + 8 = 0$ …
50Solve the equation $2x^3 - 7x + 2 = 0$ , leaving non-rational roots in the form $a \pm b\sqrt{2}$ , where $a$ and $b$ are…
51Two variables $x$ and $y$ are related by the equation $8y = \left(\frac{x}{2} - 1\right)^4$ . Given that both $x$ and $y$ …
52Factorise completely $2x^3 - 3x^2 - 5x + 6$ . Hence, solve $2e^{3y} - 3e^{2y} - 5e^y + 6 = 0$ .
53Factorise completely $x^4 - y^4$ . Answer ... Given that $x^2 - 2xy = 1 - y^2$ , express $y$ in terms of $x$ . Answer ... o…
54The equation of a polynomial is $\text{f}(x) = 2x^3 - 7x^2 + 9x - 3$ . Factorise the polynomial $\text{f}(x)$ . Determine…
55The polynomials $P(x)=ax^3+x^2+bx+10$ and $Q(x)=x^3+ax^2-5x+b$ leave the same remainder when divided by $x+2$ . Show that…
56Using long division, divide $2x^{3}-7x^{2}+2x+3$ by $2x-4$ . Solve $2x^{3}-7x^{2}+2x+3=0$ . Kun Ye claims that the solutio…
57The polynomial $\mathrm{f}(x)$ is such that the coefficient of $x^4$ is $2$ . The roots of the equation $\mathrm{f}(x) = 0$ …
58The diagram shows part of the curve $y = 16 + 6x - x^2$ intersecting the line $y = 26 - x$ at $A$ and $B$ . Find the coor…
59A polynomial $f(x)$ has a remainder of $-1$ when divided by $2x + 3$ . Find the remainder when $f(x) - 1$ is divided by $2x + 3$ …
60It is given that $\mathrm{f}(x) = Ax(e^{kx})$ , where $A$ and $k$ are constants. Find the exact values of $A$ and $k$ suc…