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MCQ Questions : The Remainder Theorem and The Factor Theorem

မှန်သော အဖြေကို ရွေးပေးရန် ဖြစ်ပါသည်။

  1. 1. The degree of a polynomial is the _____
    Explanation

    ကိန်းရှင် $x$ ပါဝင်သော polynomial တစ်ခုတွင် $x$ ၏ အကြီးဆုံးထပ်ညွှန်းကို ၎င်း polynomial ၏ degree (သို့) order ဟု ခေါ်သည်။
  2. When $9 x^{2}-6 x+2$ is divided by $x-3$, the remainder will be
    Explanation

    Let $f(x)=9 x^{2}-6 x+2$.

    When $f(x)$ is divided by $x-3$,

    The remainder = $f(3)=9 (3)^{2}-6 (3)+2=65$
  3. If a polynomial $f(x)$ is divided by a linear divisor $(x-a)$, then the remainder is
    Explanation

    Remainder Theorem ကို မေးထားခြင်း ဖြစ်ပါသည်။
  4. For what value of $k, x+4$ is a factor of the polynomial $x^{2}-x-(2 k-2) ?$
    Explanation

    $x+4$ is a factor of the polynomial $x^{2}-x-(2 k-2)$.

    By factor theorem,

    $(-4)^{2}-(-4)-(2 k-2)=0$

    $16+4-2(k-1)=0$

    $2(k-1)=20$

    $k=11$
  5. If the two roots of the equation $x^{3}-x^{2}-5 x+5=0$ are $\sqrt{5}$ and $-\sqrt{5}$, then the third root is
    Explanation

    The two roots of the equation $x^{3}-x^{2}-5 x+5=0$ are $\sqrt{5}$ and $-\sqrt{5}$.

    Let the third root be $k$.

    $\therefore \quad x^{3}-x^{2}-5 x+5=\left(x-\sqrt{5}\right)\left(x+\sqrt{5}\right)\left(x-k\right)$.

    $\therefore\quad \sqrt{5}\times \left(-\sqrt{5} \right)\times (-k)=5$

    $\quad\quad 5k=5$

    $\quad\quad k=1$
  6. What should be subtracted to the polynomial $x^{2}-16 x+30$, so that $x-15$ is the factor of the resulting polynomial?
    Explanation

    Let $f(x) = x^{2}-16 x+30$.

    When $f(x)$ is divided by $x-15$,

    The remainder = $f(15) = (15)^{2}-16 (15)+30 = 15(15 - 16 + 2) = 15$

    Hence, $15$ should be subtracted.
  7. The quadratic polynomial $a x^{2}+b x+c=0$ where $c=\dfrac{b^{2}}{4 a}$ has _____real roots.
    Explanation

    $a x^{2}+b x+c=0$ where $c=\dfrac{b^{2}}{4 a}$

    $\therefore\quad 4ac = b^2$

    $\therefore\quad b^2-4ac = 0$

    $\therefore\quad a x^{2}+b x+c=0$ has only one root.
  8. The remainder when $x^{4}-3 x^{2}+4 x+1$ is divided by $x^{2}-x-1$ is
    Explanation

  9. If the sum and the product of the roots of $x^{2}+b x+c=0$ are $3$ and $2$ , then
    Explanation

    The sum and the product of the roots of $x^{2}+b x+c=0$ are $3$ and $2$.

    Let the roots be $p$ and $q$.

    $\therefore\quad p+q = 3, \ pq=2$

    $\therefore\quad x^{2}+b x+c=(x-p)(x-q)$

    $\therefore\quad x^{2}+b x+c=x^{2}-(p+q) x+pq$

    $\therefore\quad x^{2}+b x+c=x^{2}-3x+2$

    $\therefore\quad b = -3, \ c = 2$
  10. Given that $f(x)=x^{4}+x^{2}+3 x-1$ and $g(x)=x^{2}-1$. If $f(x)=q(x) g(x)+r(x)$, which of the following is true?

    $\mathbf{I.}\ q(x)=x^2+2 \quad$ $\mathbf{II.}\ r(x)=3 x+1 \quad$ $\mathbf{III.}\ g(x)$ is a factor of $f(x)$
    Explanation



    $f(x)=x^{4}+x^{2}+3 x-1$ and $g(x)=x^{2}-1$

    $\quad\quad f(x)=q(x) g(x)+r(x)$ (given)

    $\therefore\quad f(x)=(x^2+2)(x^2-1)+3x+1$

    $\therefore\quad q(x)=x^2+2, \ r(x) = 3x+1$

  11. If $x-3$ is a factor of $x^{3}-(2+k) x^{2}+7 k$, then $k=$
    Explanation

    $x-3$ is a factor of $x^{3}-(2+k) x^{2}+7 k$.

    $\therefore\quad 3^{3}-(2+k) (3)^{2}+7 k=0$

    $\therefore\quad 27-9\left(2+k\right)+7k=0$

    $\therefore\quad k=\dfrac{9}{2} = 4.5$

  12. If $f(x)=6 x^{3}+13 x^{2}+p x+q$ is exactly divisible $2 x^{2}+7 x-4$, then
    Explanation

    $f(x)=6 x^{3}+13 x^{2}+p x+q$ is exactly divisible $2 x^{2}+7 x-4$

    $2 x^{2}+7 x-4=(2x-1)(x+4)$

    $\therefore\quad 6 \left(\dfrac{1}{2}\right)^{3}+13 \left(\dfrac{1}{2}\right)^{2}+p \left(\dfrac{1}{2}\right)+q=0 \ldots(1)$

    $\quad\quad 6 \left(-4\right)^{3}+13 \left(-4\right)^{2}+p \left(-4\right)+q=0\ldots(2)$

    $\therefore\quad p=-40,\ q=16$

    $\therefore\quad f(x)=6 x^{3}+13 x^{2}-40x+16$

    $\therefore\quad f \left(\dfrac{4}{3}\right)=6 \left(\dfrac{4}{3}\right)^{3}+13 \left(\dfrac{4}{3}\right)^{2}-40\left(\dfrac{4}{3}\right)+16$

    $\hspace{2.7cm} =\dfrac{128}{9}+\dfrac{208}{9}-\dfrac{160}{3}+16$

    $\hspace{2.7cm} =0$

    $\therefore\quad 3 x-4$ is a factor of $f(x)$.
  13. If $x^{2}+2 x-3$ is a factor of $f(x)=x^{4}+2 x^{3}-7 x^{2}+a x+b$, then
    Explanation

    $x^{2}+2 x-3$ is a factor of $f(x)=x^{4}+2 x^{3}-7 x^{2}+a x+b$

    Let $f(x) = g(x)(x^{2}+2 x-3)$

    $\therefore x^{4}+2 x^{3}-7 x^{2}+a x+b = g(x)(x+3)(x-1)$

    When $x=-3,\ (-3)^{4}+2 (-3)^{3}-7 (-3)^{2}+a (-3)+b = 0$

    When $x=1,\ (1)^{4}+2 (1)^{3}-7 (1)^{2}+a (1)+b = 0$

    $\therefore\quad 3a - b =-36 \ \ldots(1)$

    $\quad\quad a + b = 4 \ \ \quad\ldots(2)$

    $\therefore\quad a= -8, b=12$
  14. If $x^{2}+a x+b$ is divided by $x+c$, then remainder is
    Explanation

    $x^{2}+a x+b$ is divided by $x+c$, then

    the remainder $= (-c)^{2}+a (-c) +b = c^2-ac+b$
  15. If $x^{3}+9 x+5$ is divided by $x$, then the remainder is
    Explanation

    $x^{3}+9 x+5$ is divided by $x$, then

    the remainder $= (0)^{3}+9 (0)+5 = 5$
  16. If $(x-1)$ is a factor of $a x-a$, then the value of $a$ is
    Explanation

    Let $f(x) = ax - a$

    $f(1) = a - a = 0$

    $\therefore\quad x-1$ is a factor of $f(x)$ for all value of $a\in \mathbb{R}$.
  17. What is the remainder when $f(x)=x^{3}+3 x^{2}+k x+k$ is divided by $p x$ ?
    Explanation

    When $f(x)=x^{3}+3 x^{2}+k x+k$ is divided by $p x$,

    The remainder = $f(0) = k$
  18. If $x^{2}-3 x+2$ is a factor of the polynomial $f(x)$, then $f(2)=$
    Explanation

    $x^{2}-3 x+2$ is a factor of the polynomial $f(x)$.

    $x^{2}-3 x+2 = (x-1)(x-2)$

    $\therefore\quad (x-1)$ and (x-2) are factors $f(x)$.

    $\therefore\quad f(2)=0$
  19. Given that $f(x)=q(x)\left(x^{2}-4\right)+3$. What is the remainder when $f(x)$ is divided by $x-2$ ?
    Explanation

    $f(x)=q(x)\left(x^{2}-4\right)+3$.

    When $f(x)$ is divided by $x-2$,

    The remainder $f(2)=q(2)\left(2^{2}-4\right)+3 =3$
  20. If $n$ is a positive integer, what is the remainder when $5 x^{2 n+1}+10 x^{2 n}-3 x^{2 n-1}+5$ is divided by $x+1$.
    Explanation

    Let $f(x)=5 x^{2 n+1}+10 x^{2 n}-3 x^{2 n-1}+5$ where $n$ is a positive integer.

    When $f(x)$ is divided $x + 1$,

    The remainder $= f(-1)$.

    $\hspace{2.5cm} =5 (-1)^{2 n+1}+10 (-1)^{2 n}-3 (-1)^{2 n-1}+5$

    $\hspace{2.5cm} =5 (-1)^{\text{odd integer}}+10 (-1)^{\text{even integer}}-3 (-1)^{\text{odd integer}}+5$ where $n\in \mathbb{J^+}$

    $\hspace{2.5cm} =5 (-1)+10 (1)-3 (-1) + 5$

    $\hspace{2.5cm} =13$
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