Factor Theorem

Q.

Factorise $x^2-2x-8$

Q. Let $P\left(x\right)=x^2+bx+c,$ where $b$ and $c$ are integers. If $P\left(x\right)$ is a factor of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5,$ then the value of $P\left(3\right)$ is

Q.

Factors of $(42–x–x²)$ are:

1. $(x–7)(x–6)$

2. $(x+7)(x–6)$

3. $(x+7)(6–x)$

4. $(x+7)(x+6)$

Q. Given that $x^2+x-6$ is a factor of $2x^4+x^3-a{x}^2+bx+a+b-1,$ then which of the following is/are true?

1. $b=3$
2. $a=3$
3. $a=16$
4. $b=16$

Q. If $3x-1$ is a factor of the polynomial $81x^3-45x^2+3a-6$, then the value of a is ______

Q. Is $(x-1),$ a factor of $8x^4+12x^3-18x+14$ ?

1. Yes
2. Cannot be determined from given data.
3. No

Q. Find the value of a and b when the polynomial $\text{f(x)}={\text{a(x)}}^3+3{\text{x}}^2+\text{bx}–3$ is exactly divisible by $(2\text{x}+3$) and leaves a remainder $-3$ when divided by $\text{(x+2)}$.​

Q. 1) Rewrite the following statement in the form of conditional statement.

i) The square of an odd number is odd.

2) Rewrite the following statement in the form of conditional statement.

ii) You will get a sweet dish after the dinner.

3) Rewrite the following statement in the form of conditional statement.

iii) You will fail, if you will not study.

4) Rewrite the following statement in the form of conditional statement.

iv) The unit digit of an integer is $0$ or $5$ if it is divisible by $5$.

5) Rewrite the following statement in the form of conditional statement.

v) The square of a prime number in not prime.

6) Rewrite the following statement in the form of conditional statement.

vi) $2b=a+c,ifa,b\text{and}c\text{are in}A.P.$

Q.

Check whether $x+3$ is a factor of $p\left(x\right)=x^2-9$.

Q. Find the value of k, if x-1 is a factor of p(x) in each of the following cases
$p\left(x\right)={2x}^2+kx+\sqrt{2}$

Q.

Find $k$ so that $3x^4+8x^3-4kx+k$ may be divisible by $x-2$.

Q. Find the value of ${}^{\prime }{k}^{\prime }$ if $x-1$ is a factor of $4x^3+3x^2-4x+k$.

Q. The value of p for which the polynomial $x^3+4x^2-px+8$ is exactly divisible by $(x-2)$ is

1. 0
2. 3
3. 5
4. 16

Q. If polynomial $P\left(x\right)=x^2+ax+b$ has factors $(x-a)\text{and}(x-b),\text{where}a,b\in R,$ then the value of $P\left(2\right)$ is

1. $8$
2. $6$
3. $7$
4. $4$

Q. Select the correct statment.

1. $x+2$ is a factor of $x^2+3x+4$
2. $x+2$ is a factor of $2x^2+4$
3. $x+2$ is a factor of $x^4-4x^3+3x^2+5x+6$
4. $x+2$ is a factor of $x^3+3x^2+5x+6$

Q. Let $P\left(x\right)=x^2+bx+c$, where $b$ and $c$ are integer. If $P\left(x\right)$ is a factor of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5$, then

1. $P\left(x\right)=0$ has imaginary roots
2. $P\left(1\right)=4$
3. $P\left(x\right)=0$ has roots of opposite sign
4. $P\left(1\right)=6$

Q. $C_0+2C_1+3.C_2+...+(n+1)C_n=2^{n-1}(n+2).$

1. True
2. False

Q. The rational zeroes of the cubic function $f\left(x\right)=x^3-2x^2-5x+6=0$ are

1. $2,-1,3$
2. $2,-1,-3$
3. $-2,1,3$
4. $-2,1,-3$

Q. Prove by factor theorem that,
(iv) $(2x-1)\text{is a factor of}6x^3-x^2-5x+2$.

Q. Is $(x-1),$ a factor of $8x^4+12x^3-18x+14$ ?

1. No
2. Yes
3. Cannot be determined from given data.

Q. Which of the following is true regarding the polynomial $f\left(x\right)=x^3-2x^2-x+2$ ?
(Use Factor Theorem)

1. $(x-1)$ is a factor of $f\left(x\right)$
2. All of the above
3. $(x-2)$ is a factor of $f\left(x\right)$
4. $(x+2)$ is a factor of $f\left(x\right)$

Q.

Write $x^4-16x^3+86x^2-176x+105$ as the product of two quadratic polynomial If one quadratic polynomial have roots $1$ and $7$.

1. $(x^2-8x+7)(x^2-8x+15)$

2. $(x^2-8x+1)(x^2-8x+105)$

3. $(x^2-8x+21)(x^2-8x+5)$

4. $(x^2-8x+8)(x^2-8x+16)$

Q. If $(1-p)$ is a root of quadratic equation $x^2+px+1(1-p)=0$, then its roots are

1. $-1,2$
2. $-1,1$
3. $0,-1$
4. $0,1$

Q. The normal at $(a{p}^2,2ap)$ on $y^2=4ax,$ meets the curve again at $(a{q}^2,2aq)$ then

1. $p^2+pq+2=0$
2. $q^2-pq+2=0$
3. $p^2-pq+2=0$
4. $p^2-pq+1=0$

Q. What number should be added to $2{\text{x}}^3-3{\text{x}}^2+7\text{x}-8$ so that the resulting polynomial is exactly divisible by $\text{(x - 1)}$.​

Q. Let $\alpha ,\beta$ be the roots of the equation $(x-a)(x-b)=c,c\ne 0$. Then the roots of the equation $(x-\alpha )(x-\beta )+c=0$ are

1. $a,b$
2. $b,c$
3. $a,c$
4. $a+c,b+c$

Q. If $x^2-3x+2$ is a factor of $x^4-a{x}^2+b$ then the equation whose roots are $a,b$ is

1. $x^2+9x-20=0$
2. $x^2-9x-20=0$
3. $x^2+9x+20=0$
4. $x^2-9x+20=0$

Q. Which of the following is true regarding the polynomial $f\left(x\right)=x^3-2x^2-x+2$ ?
(Use Factor Theorem)

1. $(x-2)$ is a factor of $f\left(x\right)$
2. All of the above
3. $(x+2)$ is a factor of $f\left(x\right)$
4. $(x-1)$ is a factor of $f\left(x\right)$

Q.

Factorize $:2x^2-7x+5=0$

1. $(x-2)(x-\frac{7}{2})=0$

2. $(2x-7)(x-1)=0$

3. $(2x-5)(x-1)=0$

4. $(x-2)(x-\frac{7}{5})=0$

Q. If polynomial $P\left(x\right)=x^2+ax+b$ has factors $(x-a)\text{and}(x-b),\text{where}a,b\in R,$ then the value of $P\left(2\right)$ is

1. $8$
2. $7$
3. $4$
4. $6$