Factor Theorem

Q. P(x)=x^2 + 2(√2)x - 6 find the zero of the polynomial and verify between zero and coefficients

Q. Verify division algorithm for the polynomials $f\left(x\right)=8+20x+x^2-6x^3$ and $g\left(x\right)=2+5x-3x^2$

Q. Obtain all zeros of the polynomial f(x) = 2x⁴ + x³ − 14x² − 19x − 6, if two of its zeros are −2 and −1.

Q. Find all the zeros of the polynomial x⁴ + x³ − 34x² − 4x + 120, if two of its zeros are 2 and −2.

Q. If x³ + x² − ax + b is divisible by (x² − x), write the values of a and b.

Q. A class of 20 boys and 15 girls is divided into new groups to teach them in different groups so that each group has x boys and y girls . Find x, y and n

Q. What should be subtracted to the polynomial x² − 16x + 30, so that 15 is the zero of the resulting polynomial?

(a) 30
(b) 14
(c) 15
(d) 16

Q.

Explained answer of

If (x+a) is a factor of 2x²+2ax+5x+10 find a

Q.

It is being given that 1 is zeros of polynomial 7x-x³-6.find the other zeros of polynomial

Q. If x + 2 is a factor of x² + ax + 2b and a + b = 4, then

(a) a= 1, b = 3
(b) a = 3, b = 1
(c) a = −1, b = 5
(d) a = 5, b = −1

Q.

If x + a is a common factor of expressions $f\left(x\right)=x^2+px+qandg\left(x\right)=x^2+mx+n;$ show that : $a=\frac{n-q}{m-p}$

Q. What must be subtracted from $8x^4+14x^3-2x^2+7x-8$ so that the resulting polynomial is exactly divisible by $4x^2+3x-2$ [4 MARKS]

Q. Obtain all zeros of f(x) = x³ + 13x² + 32x + 20, if one of its zeros is −2.

Q. If $\left(x+a\right)$ is a factor of the polynomial $2x^2+2ax+5x+10$, find the value of a.

Q. Find the value of p, for which one root of the quadratic equation $p{x}^2-14x+8=0$ is 6 times the other.

Q.

Given the area of rectangle is A $=25a^2-35a+12$. The length is given as ($5a-3$). Therefore, the width is?

Q.

Find the value of a, if (x-a) is a factor of x³- a²x + x+ 2

Q. A polynomial g(x) of degree zero is added to the polynomial 2x^3 + 5x^2-14x+10 so that it becomes exactly divisible by 2x-3. Find g(x).

Q.

If a polynomial p(x) has a factor (x-a), then p(a)=0.

1. False

2. True

Q.

Given that, x² + 2x - 3 is a factor of x⁴ + 6x³ + 2ax² + bx - 3a. Find the values of a and b.

Q.

If a polynomial of degree 7 is divided by a polynomial of degree 3 then find the degree of quotient

Q. Find all zeros of the polynomial f(x) = 2x⁴ − 2x³ − 7x² + 3x + 6, if its two zeros are $-\sqrt{\frac{3}{2}}\mathrm{and}\sqrt{\frac{3}{2}}$.

Q. Obtain all zeros of the polynomial f(x) = x⁴ − 3x³ − x² + 9x − 6, if two of its zeros are $-\sqrt{3}\mathrm{and}\sqrt{3}$

Q.

Statement1: The polynomial $P\left(x\right)=4x^3–3x^2+5x–6$ when divided by $x–1$ gives zero as the remainder.

Statement2: $(x–1)$ is a factor of the polynomial $P\left(x\right)=4x^3–3x^2+5x–6$.

1. Both the statements are true and statement 2 is the correct explanation of statement 1.

2. Statement 1 is false and statement 2 is true.

3. Both the statements are true and statement 2 is not the correct explanation of statement 1.

4. Statement 1 is true and statement 2 is false.

Q. By actual division, show that $x^2-3$ is a factor of $2x^4+3x^3-2x^2-9x-12$

Q.

Find the values of m and n in polynomial 2x³+MX²+nx-14 such that x-1and x+2 are it's factors

Q. If a polynomial p(x) has a factor (x-a), then according to factor theorem ______.

1. p(a) = p(x)
2. p(a) = 0
3. p(x) = p(a)
4. None of the above

Q.

Find the value of 'a', if (x-a) is a factor of $x^3-a{x}^2+x+2.$

Q.

If the polynomial (2x + 3 ) is a factor of the polynomial
2x³ + 9x² - x – b. The value of b is_______

1. 10

2. 5

3. 20

4. 15

Q. If (2x+1) is a factor of 6x3 + 5x2 + ax - 2 , find the value of a .

​​x3 - cube
x2 - square