Factor Theorem

Q.

Factorize:$x^3+13x^2+32x+20$.

Q.

Find the values of a and b so that the polynomial $(x^3-10x^{2}ax+b)$ is exactly divisible by (x -1) as well as (x -2).

Q.

Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case: $\left(\mathrm{i}\right)2{\mathrm{x}}^3+{\mathrm{x}}^2-5\mathrm{x}+2;1/2,1,-2$

Q. find k so that x2+2x+k is a factor of 2x4+x3-14x2+5x+6.also find all the zeroes two polynomial

Q.

If x + 2 and x - 1 are the factors of $x^3+10x^2$ + mx + n, then the values of m and n are respectively

1. 5 and -3

2. 5 and -8

3. 7 and -18

4. 23 and -19

Q.

If both x - 2 and $x-\frac{1}{2}$ are factors of $p{x}^2$ + 5x + r, then

1. p = r

2. 2p + r = 0

3. p + r = 0

4. p + 2r = 0

Q. If (x³ + ax² + bx + 6) has x-2 as a factor and leaves a remainder 3 when divided by x-3, find the values of a and b.

Q.

The polynomial $a{x}^3+b{x}^2+x–6$ has $(x+2)$ as a factor and leaves a remainder $4$

when divided by $(x–2)$. Find $a$ and $b$.

1. 0, 2

2. 0, 4

3. 2, 4

4. 2, 2

Q.

If x + a is a factor of x² + px + q and x² + mx + n then the value of a in terms of n, q, m and p is?

Q. Question 22
If x + 1 is a factor of $a{x}^3+x^2–2x+4a–9$, then find the value of a.

Q.

Factorise the polynomial $x^4+x^3-7x^2-x+6$ using factor theorem.

Q.

If x = - $\frac{1}{2}$ is a zero of the polynomial p(x)

= $8x^3-a{x}^2$ - x + 2, find the value of a.

Q.

An unknown polynomial yields a reminder 2 upon division by x-1, and remainder 1 upon division by x-2. What remainder is obtained if this polynomial is divided by {x-1}{x-2}.

Q. Find the value of k, if x − 1 is a factor of x² + x + k.

Q. How can we factories 3x^2+2x+1

Q.

Find the value of$k$, if $x–1$is a factor of $P\left(x\right)$ in the following case: $P\left(x\right)=k{x}^2-\sqrt{2}x+1$

Q. If (x - 1) is the factor of f(x) = $x^2-ax+2$, find the value of a.

1. -2
2. -3
3. 3
4. 1

Q.

If both x + 1 and x - 1 are factors of $a{x}^3+x^2$ - 2x + b, find the values of a and b.

Q.

Find the values of a and b so that the polynomial $(x^4+a{x}^3-7x^2-8x+b)$ is exactly divisible by (x + 2) as well as (x +3).

Q.

If $x^3+p{x}^2+qx+6$ has $x-2$ as a factor and leaves remainder $3$ when divided by $x-3$ find the value of $p$ and $q$

Q. If x^2-1 is a factor of ax^4+bx^3+cx^3 +dx+e, then show that a+c+e=b+d=0.

Q.

Find the value of k if x - 3 is a factor of $k^2{x}^3-k{x}^2$ -3kx - k

Q. Find the values of p and q so that x⁴ + px³ + 2x³ − 3x + q is divisible by (x² − 1).

Q.

For what values of a is 2x³+ax^{​​​​​2}+11x+a+3 exactly divisible by 2x-1

Q.

Find the value of $k$ if $(2x-3)$ is a factor of $2x^3-9x^2+x+k.$

Q.

Using factor theorem, factorize each of the following polynomials

$x^3-6x^2$ + 3x + 10

Q.

Find the value of m for which (2x - 1) is a factor of $(8x^4+4x^3-16x^2+10x+m)$.

Q.

Find the value of a for which the polynomial $(x^4-x^3-11x^2-x+a)$ is divisible by (x +3).

Q. What must be added to $3x^3+x^2-22x+9$ so that the result is exactly divisible by $3x^2+7x-6$ ?

Q.

Check whether the polynomial p(x) = 4x³ + 4x² - x - 1 is a multiple it 2x + 1