Factor Theorem

Q. If $x+3$ is a factor of $3x^2+kx+6$, then the value of $k$ is

Q.

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

1. 

2. 

3. 

4. 

Q. If Z ^5, minus 32 can be factorised as linear factor of (Z- 2) (Z^2- pz + 4)(Z^2 - qz+4) find p^2+2q

Q.

Solve the given expansion.
$(1+2x-3x^2{)}^4$

Q.

If $\sqrt{5}{x}^2+8x+3\sqrt{5}=0$ is factorized into the form $\sqrt{5}{x}^2+px+qx+3\sqrt{5}=0$, then what is the value of $p^2+q^2$ ?

1. $34$

2. $9$

3. $16$

4. $25$

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

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

Q. $\underset{x\to -2}{\lim }\frac{x^3+x^2+4x+12}{x^3-3x+2}$

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

Q. Which of the following is/are true?

1. $x+2$ is a factor of $x^3+3x^2+5x+6$
2. $x+2$ is not a factor of $x^3+3x^2+5x+6$
3. $x+2$ is not a factor of $2x+4$
4. $x+2$ is a factor of $2x+4$

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

Q. Let $P\left(x\right)=x^2+bx+c$, where $b$ and $c$ are integers. If $P\left(x\right)$ is a facter of both $x^4+6x^2+25$ and $3x^4+4x^2+28x+5$, then value of $P\left(1\right)$ is -

1. $12$
2. $4$
3. $10$
4. $8$

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

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

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

Q. If $x^2-1$ is a factor of $x^4+a{x}^3+3x-b$, then

1. $a=3,b=1$
2. None of these
3. $a=3,b=–1$
4. $a=–3,b=1$

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

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

Q.

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

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

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

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

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

Q.

If $(x+1)$ is a factor of $x^4-(p-3)x^3-(3p-5)x^2+(2p-7)x+6$, then $p=$

1. $4$

2. $1$

3. $2$

4. None of these