Zeros of a Polynomial

Q.

If $\alpha$ and $\beta$ are the zeros of the polynomial $f\left(x\right)=x^2+x-2$, find the value of $(\frac{1}{\alpha }-\frac{1}{\beta })$.

Q.

What's the formula for finding a cubic polynomial given the sum, sum taken to at a time, and the product of its zeroes ?

Q. Which of the following is not a zero of the polynomial $p\left(x\right)=x^3-11x^2+36x-36?$

1. 2
2. 3
3. 4
4. 6

Q.

If one zero of the quadratic polynomial $k{x}^2+3x+k$ is 2 then find the value of k.

Q.

For each of the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorization.

$\left(-2\sqrt{3}\right),\left(-9\right)$

Q. Find the cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and product of its zeros as 3, −1 and − 3 respectively.

Q.

If one of the zeros of the quadratic polynomial $\left(k-1\right)x^2+kx+1$ is $-3$, then find the value of $k$.

Q.

One of the factors of $(25x^2-1)+(1+5x{)}^2$ is

A) $5+x$
B) $5-x$
C) $5x-1$
​​​​​​​D) $10x$

Q.

Which of the following statements is correct?

1. If ${\mathrm{x}}^6+1$ is divisible by $\left(\mathrm{x}+1\right)$, then the remainder is $-2$

2. If ${\mathrm{x}}^6+1$ is divisible by $\left(\mathrm{x}-1\right)$, then the remainder is $2$

3. If ${\mathrm{x}}^6+1$ is divisible by $\left(\mathrm{x}+1\right)$, then the remainder is $1$

4. If ${\mathrm{x}}^6+1$ is divisible by $\left(\mathrm{x}-1\right)$, then the remainder is $-1$

Q.

$\text{Find the zeroes of the polynomial}{x}^2-2.$

1. $\sqrt{2}\text{and}-\sqrt{5}$

2. $4\text{and}3$

3. $\sqrt{2}\text{and}-\sqrt{2}$

4. $-5\text{and}2$

Q.

It is given that -1 is one of the zeros of the polynomial $x^3+2x^2-11x-12$. Find all the zeros of the given polynomial.

Q.

If $(a − b)$,$a$ and $(a+b)$ are zeros of the polynomial $2x^3-6x^2+5x-7$, write the value of $a$.

Q. The product of the zeros of x³ + 4x² + x − 6 is

(a) −4
(b) 4
(c) 6
(d) −6

Q.

Write the zeroes of the polynomial $x^2+2x+1$.

Q. Question 4
The number of polynomials having zeroes as –2 and 5 is
(A) 1
(B) 2
(C) 3
(D) more than 3

Q. If two zeroes of the polynomial x³ + x² − 9x − 9 are 3 and −3, then its third zero is

(a) −1
(b) 1
(c) −9
(d) 9

Q.

If 3 is a zero of the polynomial $2x^2+x+k$, find the value of k.

Q. Find a cubic polynomial whose zeroes are 2, −3 and 4

Q. If $\sqrt{5}\mathrm{and}-\sqrt{5}$ are two zeroes of the polynomial x³ + 3x² − 5x − 15, then its third zero is

(a) 3
(b) −3
(c) 5
(d) −5

Q. $\text{A quadratic polynomial whose zeroes}\text{are}\frac{3}{4}\text{and}\frac{1}{2}\text{is}$

1. $\frac{8x^2-10x+3}{8}$
2. $\frac{10x^2-8x+3}{4}$
3. $\frac{8x^2+10x+3}{8}$
4. $\frac{10x^2+8x+3}{4}$

Q.

$\text{Find the zeros of the quadratic}\text{polynomial}{x}^2-8x+12.$

1. $2\text{and}6$

2. $8\text{and}4$

3. $10\text{and}2$

4. $4\text{and}5$

Q.

What will the quotient and remainder be on division of $a{x}^2+bx+c$ by $p{x}^3+q{x}^2+rx+s,p\ne 0$ ?

Q. If two of zeroes of the polynomial $p\left(x\right)=2x^4+7x^3-19x^2-14x+30are\sqrt{2}and-\sqrt{2},$ then the other zeroes are

1. -2, -3
2. $\frac{3}{2},-5$
3. $\sqrt{3},-\sqrt{3}$
4. $\frac{5}{2},-3$

Q.

Find the quadratic polynomial whose zeroes are $\frac{6+\sqrt{3}}{3}\text{and}\frac{6-\sqrt{3}}{3}.$

1. $x^2-16x+9$

2. $3x^2-12x+11$

3. $6x^2-4x+1$

4. $7x^2-4x+11$

Q. If the sum of zeroes of a cubic polynomial is -3 and product of the zeroes is 2 what is the third zero?

Q.

If -4 is a zero of the polynomial $x^2-x-(2k+2)$ then find the value of k.

Q.

Question 78

A car can travel 240km in 15 litres of petrol. How much distance will it travel in 25 litres of petrol?

Q. Question 1 (i)
For each of the following, find a quadratic polynomial whose sum and product of the zeroes respectively are as given. Also, find the zeroes of these polynomials by factorization.
i) $\frac{-8}{3},\frac{4}{3}$

Q.

Obtain zeroes of 4√3x² +5x - 2√3 and verify relation between its zeroes and coefficients

Q.

Find the zero of the polynomials in the following case: $p\left(x\right)=cx+d,c\ne 0,c,d$ are real numbers.