Sample QuestionsFactorization Of Polynomials questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $x+1$ is a factor of the polynomial $2 x^2+k x$, then $k=$
Answer: D.
View full solution →If both $x-2$ and $x-\frac{1}{2}$ are factor of $p x^2+5 x+r$, then
- ✓
$p=r$
- B
$p+r=0$
- C
$2 p+r=0$
- D
$p+2 r=0$
Answer: A.
View full solution →If $x-3$ is a factor of $x^2-a x-15$, then $a=$
Answer: A.
View full solution →If $x^3+6 x^2+4 x+k$ is exactly divisible by $x+2$, then $k$
Answer: C.
View full solution →If $x-a$ is a factor of $x^3-3 x^2 a+2 a^2 x+b$, then the value of $b$ is:
Answer: A.
View full solution →Statement-1 (A): If $x+2 a$ is a factor of $f(x)=x^5-4 a^2 x^3+2 x+2 a+3$, then $2 a-3=0$
Statement-2 (R): If $f(x)$ is divisible by $(a x+b)$, then $f\left(-\frac{b}{a}\right)=0$.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-5
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
Answer: A.
View full solution →Statement-1 (A): If $x+1$ is a factor of $f(x)=p x^2+5 x+r$, then $p+r+5=0$.
Statement-2 (R): If $x-2$ and $2 x-1$ are factors of $f(x)=p x^2+5 x+r$, then $p=r$.
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-4
- C
Statement-1 is true, Statement-2 is false.
- ✓
Statement-1 is false, Statement-2 is true.
Answer: D.
View full solution →Statement-1 (A): If $f(x+2)=2 x^2+7 x+5$, then the remainder when $f(x)$ is divided by $(x-$ $1)$ is 0 .
Statement-2 (R): If a polynomial $f(x)$ is divided by $(a x+b)$, then the remainder is $f(b / a)$.
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-3
- ✓
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
Answer: C.
View full solution →Statement-1 (A): If the polynomial $f(x)=3 x^4-11 x^2+6 x+k$ when divided by $(x-3)$ leaves remainder 7 , then $k=-155$.
Statement-2 $( R )$ : If a polynomial is divided by $(x-a)$, the remainder is $f(a)$.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- B
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-2
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
Answer: A.
View full solution →Statement-1 (A): If the polynomial $p(x)=x^3+a x^2-2 x+a+4$ has $(x+a)$ as one of its factors, then $a=-\frac{4}{3}$.
Statement-2 (R): If $f(x)=a x^2+b+c$ is exactly divisible by $2 x-3$ then $4 a+6 b+9 c=0$
- A
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- ✓
Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1
- C
Statement-1 is true, Statement-2 is false.
- D
Statement-1 is false, Statement-2 is true.
Answer: B.
View full solution →The remainder when $f(x)=x^{45}$ is divided by $x^2-1$ is _______________ .
View full solution →If the polynomial $f(x)=5 x^5-3 x^3+2 x^2-k$ gives remainder 1 when divided by $x+1$, then $k=$ _______________ .
View full solution →If $p(x)=x^2-4 x+3$, then $p(2)-p(-1)+p\left(\frac{1}{2}\right)=$ _______________ .
View full solution →The remainder when $x^{15}$ is divided by $x+1$ is _______________ .
View full solution →The degree of a polynomial $f(x)$ is 7 and that of polynomial $f(x) g(x)$ is 56 , then degree of $g(x)$ is $\qquad$ . _______________ .
View full solution →Identify constant, linear, quadratic and cubic polynomial from the following polynomials:
$r(x)=3 x^3+4 x^2+5 x-7$
View full solution →Write the coefficients of $x^2$ in the following:
$9-12 x+x^3$
View full solution →Identify the polynomials in the following: $\text{p(x)}=\frac{2}{3}\text{x}^2-\frac{7}{4}\text{x}+9$
View full solution →Identify the polynomials in the following: $\text{f(x)}=2+\frac{3}{\text{x}}+4\text{x}$
View full solution →Classify the following polynomials as linear, quadratic, cubic and biquadratic polynomials: $3x - 2$
View full solution →What must be subtracted from $x^3 - 6x^2 - 15x + 80$ so that the result is exactly divisible by $x^2 + x - 12?$
View full solution →Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following case: $f(x)=x^2, x=0$
View full solution →Find the remainder when $x^3+3 x^3+3 x+1$ is divided by:
$x$
View full solution →In the following, use factor theorem to find whether polynomial $g(x)$ is a factor of polynomial $f(x)$ or, not: $f(x)=3 x^4+17 x^3+9 x^2-$ $7 x-10 ; g(x)=x+5$
View full solution →If $x+1$ is a factor of $x^3+a$, then write the value of $a$.
View full solution →Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following case: $\text{f(x)}=2\text{(x)}+1,\text{x}=\frac{1}{2}$
View full solution →Verify whether the indicated numbers are zeros of the polynomials corresponding to them in the following case:
$f(x) = x^2 - 1, x = 1, -1$
View full solution →Find the value of a such that $(x - 4)$ is a factors of $5x^3 - 7x^2 - ax - 28.$
View full solution →Find the remainder when $x^3 + 3x^3 + 3x + 1$ is divided by: $\text{x}+\pi$
View full solution →Factorize the following polynomials: $4x^3 + 20x^2 + 33x + 18$ given that $2x + 3$ is a factor.
View full solution →If $(x + y)^3 - (x - y)^3 - 6y(x^2 - y^2) = ky^2$, then $k =$
Answer: D.
View full solution →The expression $(a - b)^3 + (b - c)^3 + (c - a)^3$ can be factorized as:
Answer: B.
View full solution →The value of $\frac{(2.3)^3-0.027}{(2.3)^2+0.69+0.09},$ is:
Answer: A.
View full solution →The factors of $x^3 - 7x + 6$ are:
- A
$x(x - 6)(x - 1)$
- B
$(x^2 - 6)(x - 1)$
- C
$(x + 1)(x + 2)(x + 3)$
- ✓
$(x - 1)(x + 3)(x - 2)$
Answer: D.
View full solution →The expression $x^4 + 4$ can be factorized as:
- ✓
$(x^2 + 2x + 2)(x^2 - 2x + 2)$
- B
$(x^2 + 2x + 2)(x^2 + 2x - 2)$
- C
$(x^2 - 2x - 2)(x^2- 2x + 2)$
- D
$(x^2 + 2)(x^2 - 2)$
Answer: A.
View full solution →In the following, using the remainder theorem, find the remainder when $f(x)$ is divided by $g(x)$ and verify the by actual division: $f(x) = 9x^3 - 3x^2 + x - 5$, $\text{g(x)}=\text{x}-\frac{2}{3}$
View full solution →Find the value of a, if $x + 2$ is a factor of $4x^4 + 2x^3 - 3x^2 + 8x + 5a.$
View full solution → Using factor theorem, factorize the following polynomials: $2y^3 + y^2 - 2y - 1$
View full solution →If $\text{x}=-\frac{1}{2}$ is zero of the polynomial $p(x) = 8x^3 - ax^2 - x + 2$, Find the value of a.
View full solution →In the following, using the remainder theorem, find the remainder when $f(x)$ is divided by $g(x)$ and verify the by actual division: $f(x) = x^3 + 4x^2 - 3x + 10, g(x) = x + 4$
View full solution →