Relationship between Zeroes and Coefficients of a Polynomial

Q.

If $\alpha$ and $\beta$ are the zeros of the polynomial $f\left(x\right)=6x^2+x-2$, find the value of $(\frac{\alpha }{\beta }+\frac{\beta }{\alpha }).$

Q.

If $\alpha ,\beta$ be the zeros of the polynomial $2x^2+5x+k$ such that ${\alpha }^2+{\beta }^2+\alpha \beta =\frac{21}{4}$ then k = ?

(a) 3 (b) -3 (c) -2 (d) 2

Q. If one zero of the quadratic polynomial, $f\left(x\right)=4x^2-8kx-9$ is negative of the other, then the value of k is

1. 0
2. 
   1
3. 
   2
4. 
   3

Q.

Find a quadratic polynomial whose zeroes are $-1\text{and}3$ . Verify the relation between the coefficient and zeroes of the polynomial.

Q.

If $\alpha and\beta$ are the zeros of $x^2+5x+8$ then the value of $(\alpha +\beta )$ is

(a) 5 (b) -5 (c) 8 (d) -8

Q.

If $\alpha and\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=x^2-3x-2$, find a quadratic polynomial whose zeros are $\frac{1}{2\alpha +\beta }and\frac{1}{2\beta +\alpha }$

Q.

If $\alpha ,\beta$ are the zeros of the polynomial $x^2+6x+2$ then $(\frac{1}{\alpha }+\frac{1}{\beta })$ = ?

(a) 3 (b) -3 (c) 12 (d) -12

Q.

If $\alpha$ and $\beta$ are the zeros of the polynomial $f\left(x\right)=5x^2-7x+1$, find the value of $(\frac{1}{\alpha }+\frac{1}{\beta }).$.

Q.

If $\alpha ,\beta$ are the zeroes of the polynomial $x^2-px+36$ and ${\alpha }^2$+${\beta }^2$ = 9, then what is the value of p?

1. ±6

2. ±3

3. ±8

4. ±9

Q. If 𝜶 and 𝜷 are the zeroes of the polynomial $6y^2-7y+2$, find a quadratic polynomial whose zeroes are $\frac{1}{\alpha }$ , $\frac{1}{\beta }$.

Q.

If the product of the zeros of the quadratic polynomial $x^2-4x+k$ is 3 then write the value of k.

Q. If one of the zero of x²-3kx+4k is twice the other. Find the value of k.

Q.

If two of the zeros of the cubic polynomial $a{x}^3+b{x}^2+cx+d$ are 0 then the third zero is

(a) $\frac{-b}{a}$ (b) $\frac{b}{a}$ (c) $\frac{c}{a}$ (d) $\frac{-d}{a}$

Q.

Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and the product of its zeros as 5, -2 and -24 respectively.

Q.

If the sum of the roots of the equation $a{x}^2+2x+3a=0$ is equal to their product, then value of $a$ is:

1. $-\frac{2}{3}$

2. $-3$

3. $4$

4. $-\frac{1}{2}$

Q.

If $\alpha and\beta$ are the zeros of $2x^2+5x-9$ then the value of $\alpha \beta$ is

(a) $\frac{-5}{2}$ (b) $\frac{5}{2}$ (c)$\frac{-9}{2}$ (d)$\frac{9}{2}$

Q. If the zeroes of the polynomial $x^3-3x^2+x+1$ are $a-b,a,a+b,$ find $a$ and $b$

Q.

If $\alpha ,\beta ,\gamma$ are the zeros of the polynomial $x^3-6x^2-x+30$ then $(\alpha \beta +\beta \gamma +\gamma \alpha )$ = ?

(a) -1 (b) 1 (c) -5 (d) 30

Q.

If $\alpha$ and $\beta$ are the zeros of the polynomial $2x^2+7x+5$, write the value of $\alpha +\beta +\alpha \beta .$

Q.

The sum of the zeros and the product of zeros of a quadratic polynomial are $\frac{-1}{2}and-3$ respectively. Write the polynomial.

Q.

Verify that 5, -2 and $\frac{1}{3}$ are the zeros of the cubic polynomial $p\left(x\right)=3x^3-10x^2-27x+10$ and verify the relation between its zeros and coefficients.

Q.

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

(a) $\frac{5}{6}$ (b) $\frac{-5}{6}$ (c) $\frac{6}{5}$ (d) $\frac{-6}{5}$

Q. $\alpha and\beta \text{are the zeroes of the quadratic}\text{polynomial},f\left(x\right)=x^2-5x+4.\text{Find out the value of}\frac{1}{\alpha }+\frac{1}{\beta }-2\alpha \beta ?$

1. 
   $-27/4$
2. 
   $27/4$
   
3. 
   $25/2$
4. 
   $-25/2$

Q.

Question 1 (iv)
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

$4u^2+8u$

Q.

Verify whether the following are zeroes of the polynomial, indicated against them: $\mathrm{p}\left(\mathrm{x}\right)=\mathrm{lx}+\mathrm{m},\mathrm{x}=\frac{-\mathrm{m}}{\mathrm{l}}$

Q. Find the value of $\frac{1}{\alpha }+\frac{1}{\beta }+\frac{1}{\gamma }$ for the polynomial $x^3+2x-5$.

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

Q. If -2 and -4 are the zeroes of a quadratic polynomial whose leading coefficient is 2, then find the value of (product of zeroes $\times$ coefficient of x).

[Note: In a polynomial, the term containing the highest power of x (i.e. a_nxⁿ) is the leading term, and we call a_n the leading coefficient.]

1. 96
2. -96
3. 16
4. -16

Q. Write the coefficient of the polynomial p(z) = z⁵ − 2z² + 4.

Q.

Find the zeroes of the polynomial xsquare -3 and verify the relationship between the zeroes and the coefficients

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

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