Factor Theorem

Q.

Let $P\left(x\right)$ be a real polynomial of degree $3$ which vanishes at $x=-3.$ Let $P\left(x\right)$ have local minima at $x=1$, local maxima at$x=-1$ and ${\int }_{-1}^{1}P\left(x\right)dx=18$, then the sum of all the coefficients of the polynomial $P\left(x\right)$ is equal to

Q.

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

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

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

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

4. $(2x-5)(x-1)=0$

Q. The absolute difference of the maximum and the minimum value of $x$ satisfying $x^4-16x^3+86x^2-176x+105=0$ is

Q.

If $x^2$ - 6x + 11 is a factor of bi-quadratic equation $x^4$ + $x^3$- 25 $x^2$ + 41x + 66 = 0. Find the product of other factor of bi-quadratic equation and x+1.

1. $x^3+x^2+x+6$

2. $x^3+8x^2+13x+6$

3. $x^3+7x^2+5x+1$

4. $x^2+7x+6$

Q. Let $P\left(x\right)$ be a quadratic polynomial with real coefficients satisfying $x^2-2x+2\le P\left(x\right)\le 2x^2-4x+3$ for all $x\in R.$ Suppose $P\left(7\right)=64,$ then the value of $P\left(13\right)$ is

1. $242$
2. $252$
3. $253$
4. $201$

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. None of these

2. $4$

3. $2$

4. $1$

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.

If $x^2$ - 6x + 11 is a factor of bi-quadratic equation $x^4$ + $x^3$- 25 $x^2$ + 41x + 66 = 0. Find the product of other factor of bi-quadratic equation and x+1.

1. $x^3+8x^2+13x+6$

2. $x^2+7x+6$

3. $x^3+7x^2+5x+1$

4. $x^3+x^2+x+6$

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. $6$
4. $4$

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

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

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

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

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

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

Q.

If a, b, c are real and $x^3-3b^{2}x+2c^3$ is divisible by $x-a$ and $x-b$, then

1. a = b = c, a = -2b = -2c

2. a = -b = -c

3. a = 2b = 2c

4. a=b=c , a=2b=2c

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. If $x^2-3x+2$ be a factor of $x^4-p{x}^2+q$, then $(p,q)=$

1. $(5,4)$
2. $(4,3)$
3. $(4,5)$
4. $(3,4)$

Q.

If the equation $2x^4+7x^3+ax+b=0$ has four real roots. Then find the value of a and b. Given that (x - 3) and (x - 1) may exactly divide the above given expression.

1. a = 6, b = - 11

2. a = 11, b = - 6

3. a = 12, b = - 5

4. a = 12, b = 5

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

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

Q.

If $x^2-3x+2$ be a factor of $x^4-p{x}^2+q$, then (p, q) =

1. (4, 3)

2. (4, 5)

3. (5, 4)

4. (3, 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. $9$

2. $34$

3. $25$

4. $16$

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

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

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

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

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

Q. If $x^2-3x+2$ be a factor of $x^4-p{x}^2+q$, then $(p,q)=$

1. $(3,4)$
2. $(4,5)$
3. $(5,4)$
4. $(4,3)$

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. $(x-1)$ is a factor of $f\left(x\right)$
4. $(x-2)$ is a factor of $f\left(x\right)$

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. $a,c$
4. $a,b$

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. $6$
4. $8$

Q.

If $x^3-18x-35=(x-5).p\left(x\right)$.
Find polynimial $p\left(x\right)$.

1. $x^2+2x+3$

2. $x^2+3x+2$

3. $x^2+7x+5$

4. $x^2+5x+7$