Practice Problems : The Remainder and Factor Theorem

1.        What number should be added to $ \displaystyle 2x^3 - 3x^2 - 8x$ so that the resulting polynomial leaves the remainder $ \displaystyle 10$ when divided by $ \displaystyle 2x + 1$?

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Let the number to be added be $ \displaystyle k$ and the resulting polynomial be $ \displaystyle f(x)$.

$ \displaystyle \therefore \ f(x)=2{{x}^{3}}-3{{x}^{2}}-8x+k$

When $ \displaystyle f(x)$ is divided by $ \displaystyle 2x+1$, the remainder is 10.

$ \displaystyle \begin{array}{l}\therefore \ f\left( {-\displaystyle \frac{1}{2}} \right)=10\\\\\ \ 2{{\left( {-\displaystyle \frac{1}{2}} \right)}^{3}}-3{{\left( {-\displaystyle \frac{1}{2}} \right)}^{2}}-8\left( {-\displaystyle \frac{1}{2}} \right)+k=10\\\\\therefore \ -\displaystyle \frac{1}{4}-\displaystyle \frac{3}{4}+4+k=10\\\\\therefore \ k=7\end{array}$

Hence, the number to be added is 10.

2.        What number should be subtracted from $ \displaystyle 6x^3 + 7x^2 - 9x+12$ so that $ \displaystyle 3x - 1$ is the factor of the resulting polynomial?

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Let the number to be subtracted be $ \displaystyle k$ and the resulting polynomial be $ \displaystyle f(x)$.

$ \displaystyle \therefore \ f(x)=6x^3 + 7x^2 - 9x+12-k$

Since $ \displaystyle 3x - 1$ is the factor of $ \displaystyle f(x)$,

$ \displaystyle \begin{array}{l}\ \ f\left( {\displaystyle \frac{1}{3}} \right)=0\\\\\ \ 6{{\left( {\displaystyle \frac{1}{3}} \right)}^{3}}+7{{\left( {\displaystyle \frac{1}{3}} \right)}^{2}}-9\left( {\displaystyle \frac{1}{3}} \right)+12-k=0\\\\\therefore \ \displaystyle \frac{2}{9}+\displaystyle \frac{7}{9}-3+12-k=0\\\\\therefore \ k=10\end{array}$

Hence, the number to be subtracted is 10.

3.        When divided by $ \displaystyle x - 3$ the polynomials $ \displaystyle x^3 - px^2 + x + 6$ and $ \displaystyle 2x^3 - x^2 - (p + 3) x - 6$ leave the same remainder. Find the value of $ \displaystyle p$.

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Let $ \displaystyle f(x)={{x}^{3}}-p{{x}^{2}}+x+6$ and $ \displaystyle g(x)=2x^3 - x^2 - (p + 3) x - 6$

$ \displaystyle f(x)$ and $ \displaystyle g(x)$ leave the same remainder when divided by $ \displaystyle x - 3$,

$ \displaystyle \begin{array}{l}\therefore \ f(3)=g(3)\\\\\ \ \ {{(3)}^{3}}-p{{(3)}^{2}}+(3)+6=2{{(3)}^{3}}-{{(3)}^{2}}-(p+3)(3)-6\\\\\ \ \ 27-9p+9=54-9-3p-9-6\\\ \ \\\therefore \ 36-9p=30-3p\ \\\\\therefore \ p=1\ \end{array}$


4.       Using remainder theorem, find the value of $ \displaystyle a$ if the division of $ \displaystyle x^3 + 5x^2 - ax + 6$ by $ \displaystyle x -1$ leaves the remainder $ \displaystyle 2a$.

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$ \displaystyle \begin{array}{l}\ \ \ \text{Let}\ f(x)={{x}^{3}}+5{{x}^{2}}-ax+6\\\\\ \ \ \text{When}\ f(x)\ \text{is divided by }x-1,\ \\\\\ \ \ \text{the remainder}\ =2a\\\\\therefore \ f(1)=2a\\\\\ \ \ {{1}^{3}}+5{{(1)}^{2}}-a(1)+6=2a\\\\\ \ \ 1+5-a+6=2a\\\\\therefore \ 3a=12\\\\\ \ \ a=4\ \ \ \end{array}$


5.        Find the value of the constants $ \displaystyle a$ and $ \displaystyle b$, if $ \displaystyle x - 2$ and $ \displaystyle x + 3$ are both factors of the expression $ \displaystyle x^3 + ax^2 + bx - 12$.

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$ \displaystyle \begin{array}{l}\ \ \ \text{Let}\ f(x)={{x}^{3}}+a{{x}^{2}}+bx-12\\\\\ \ \ (x-2)\ \text{and}\ (x+3)\ \text{are factors of }f(x).\ \\\\\therefore \ f(2)=0\\\\\ \ \ {{(2)}^{3}}+a{{(2)}^{2}}+b(2)-12=0\\\\\therefore \ \ 8+4a+2b-12=0\\\\\therefore \ 2a+b=2---(1)\\\\\ \ \text{Again}\ f(-3)=0\\\\\ \ \ {{(-3)}^{3}}+a{{(-3)}^{2}}+b(-3)-12=0\\\\\therefore \ \ -27+9a-3b-12=0\\\\\therefore \ \ 3a-b=13---(2)\\\\\ \ \ (1)+(2)\Rightarrow 5a=15\\\\\therefore \ \ a=3\\\\\therefore \ 2\left( 3 \right)+b=2\\\\\therefore \ b=-4\end{array}$


6.        If $ \displaystyle x + 2$ and $ \displaystyle x - 3$ are factors of $ \displaystyle x^3 + ax + b$, find the values of $ \displaystyle a$ and $ \displaystyle b$. With these values of $ \displaystyle a$ and $ \displaystyle b$, factorise the given expression.

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$ \displaystyle \begin{array}{l}\ \ \ \text{Let}\ f(x)={{x}^{3}}+ax+b\\\\\ \ \ (x+2)\ \text{and}\ (x-3)\ \text{are factors of }f(x).\ \\\\\therefore \ f(-2)=0\\\\\ \ \ {{(-2)}^{3}}+a(-2)+b=0\\\\\therefore \ \ -8-2a+b=0\\\\\therefore \ -2a+b=8---(1)\\\\\ \ \text{Again}\ f(3)=0\\\\\ \ \ {{(3)}^{3}}+a(3)+b=0\\\\\therefore \ \ 27+3a+b=0\\\\\therefore \ \ 3a+b=-27---(2)\\\\\ \ \ (2)-(1)\Rightarrow 5a=-35\\\\\therefore \ \ a=-7\\\\\therefore \ \ -2\left( {-7} \right)+b=8\\\\\therefore \ \ b=-6\\\\\therefore \ \ f(x)={{x}^{3}}-7x-6\\\\\ \ \ \ \text{Let}\ {{x}^{3}}-7x-6=(x+2)(x-3)(x+k)\\\\\therefore \ \ {{x}^{3}}-7x-6={{x}^{3}}+\left( {k-1} \right){{x}^{2}}-\left( {k+6} \right)x-6k\\\\\therefore \ k-1=0\Rightarrow k=1\\\\\therefore \ \ f(x)=(x+2)(x-3)(x+1)\\\ \ \end{array}$


7.        Given that $ \displaystyle x - 2$ is a factor of the expression $ \displaystyle x^3 + ax^2 + bx + 6$. When this expression is divided by $ \displaystyle x - 3$, it leaves the remainder 3. Find the values of $ \displaystyle a$ and $ \displaystyle b$.

8.        If $ \displaystyle x - 2$ is a factor of the expression $ \displaystyle 2x^3 + ax^2 + bx - 14$ and when the expression is divided by $ \displaystyle x - 3$, it leaves a remainder $ \displaystyle 52$, find the values of $ \displaystyle a$ and $ \displaystyle b$.

9.        If $ \displaystyle ax^3 + 3x^2 + bx - 3$ has a factor$ \displaystyle 2x + 3$ and leaves remainder -3 when divided by (x + 2), find the values of $ \displaystyle a$ and $ \displaystyle b$. With these values of $ \displaystyle a$ and $ \displaystyle b$, factorise the given expression.

10.      Given $ \displaystyle f (x) = ax^2 + bx + 2$ and $ \displaystyle g (x) = bx^2 + ax + 1$. If $ \displaystyle x - 2$ is a factor of $ \displaystyle f (x)$ but leaves the remainder $ \displaystyle -15$ when it divides $ \displaystyle g (x)$, find the values of $ \displaystyle a$ and $ \displaystyle b$. With these values of $ \displaystyle a$ and $ \displaystyle b$, factorise the expression $ \displaystyle f (x) + g (x) + 4x^2 + 7x$.

11.     When $ \displaystyle x^3 - 2x^2 + px - q$ is divided by $ \displaystyle x^2 - 2x - 3$, the remainder is $ \displaystyle x - 6$, What are the values of $ \displaystyle p$ and $ \displaystyle q$ respectively ?

12.      If $ \displaystyle x + k$ is a common factor of $ \displaystyle x^2 + px + q$ and $ \displaystyle x^2 + lx + m$, Find the value of $ \displaystyle k$ in terms of $ \displaystyle p, q, l$ and $ \displaystyle m$.

13.      When a polynomial $ \displaystyle f(x)$ is divided by $ \displaystyle x - 3$ and $ \displaystyle x + 6$, the respective remainders are $ \displaystyle 7$ and $ \displaystyle 22$. What is the remainder when $ \displaystyle f(x)$ is divided by $ \displaystyle (x - 3) (x + 6)$ ?

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$ \displaystyle \begin{array}{l}\ \ \ \text{Let}\ f(x)=Q(x)(x-3)(x+6)+ax+b\\\\\ \ \text{where is }Q(x)\ \text{a quotient and }ax+b\ \text{is }\\\ \ \text{the remainder when }f(x)\ \text{is divided by}\ (x-3)(x+6).\\\\\ \ \ \text{By the problem,}\\\\\ \ \ f(3)=7\\\\\ \ \ 3a+b=7\ ---(1)\\\\\ \ f(-6)=22\\\\\ \ \ -6a+b=22\ ---(2)\\\\\ \ \ (1)-(2)\Rightarrow 9a=-\displaystyle \frac{5}{3}\\\\\therefore \ 3\left( {-\displaystyle \frac{5}{3}} \right)+b=7\\\\\ \ \ b=12\\\\\therefore \ \text{The remainder when }f(x)\ \text{is divided by}\ \\\ \ \ (x-3)(x+6)\ \text{is}\ -\displaystyle \frac{5}{3}x+12.\end{array}$

$ \displaystyle (x-3)(x+6)$ သည္ polynomial of second degree ျဖစ္ပါသည္။

$ \displaystyle f(x)$ ကို $ \displaystyle (x-3)(x+6)$ ႏွင့္ စားေသာ အႂကြင္းသည္ စားကိန္းေအာက္ တစ္ထပ္ေလ်ာ့ပါမည္။။

ထို႔ေၾကာင့္ အႂကြင္းသည္ $ \displaystyle ax + b$ ပံုစံျဖစ္ပါမည္။

14.      Given that $ \displaystyle f (x) = x^3 + ax^2 + bx + c$. If $ \displaystyle f (1) = f(2) = 0$ and $ \displaystyle f(4) = f(0)$, find $ \displaystyle a, b$ and $ \displaystyle c$.

15.      If $ \displaystyle x – 1$ is a factor of $ \displaystyle Ax^3 + Bx^2 - 36x + 22$ and $ \displaystyle 2^B = 64^A$, find $ \displaystyle A$ and $ \displaystyle B$.

16.      When k is subtracted from $ \displaystyle 27x^3 - 9x^2 - 6x - 5$, it is exactly divisible by $ \displaystyle 3x - 1$, find $ \displaystyle k$.

17.      If $ \displaystyle x^3 + px + q$ and $ \displaystyle x^3 + qx + p$ have a common factor, show that $ \displaystyle p + q +1 = 0$.

18.      Given that $ \displaystyle f(x)=4x^3-4kx^2-x+k$, $ \displaystyle g(x)=3x^2+(1-3k)x-k$ and $ \displaystyle h(x)=f(x)+g(x)$. If $ \displaystyle x-2$ is a factor of $ \displaystyle h(x)$, find the value of If $ \displaystyle k$ and hence solve the equation $ \displaystyle h(x)=0$.

19.      What must be added to $ \displaystyle 6x^5 + 5x^4 + 11x^3 - 3x^2 + x + 1$, so that the polynomial so obtained is exactly divisible by $ \displaystyle 3x^2 - 2x + 4$?

20.      Determine the value of $ \displaystyle p$ for which the polynomial $ \displaystyle 5x^3 - x^2 + 4x+ p$ is divisible by $ \displaystyle 1 - 5x$.