Relationship Between Zeroes and Coefficients of a Quadratic Polynomial

Q.

The number of polynomials having zeroes as $-2$and $5$ is

1. $1$

2. $2$

3. $3$

4. more than $3$

Q.

Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. $\frac{1}{4},-1$

Q. $\text{Find the sum and product of roots of the}\text{polynomial}2x^2+x-5=0.$

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

Q. Find the quadratic polynomial, sum and product of whose zeroes are 0 and -1 respectively.

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

Q. If ‘α’ and ‘β’ are the zeroes of the quadratic polynomial $p\left(x\right)=x^2+7x+5,\text{then}(\alpha -\beta {)}^2$ equals

Q.

If $2$ zero's of a cubic polynomial are $0$, then it has no first degree or constant terms.

1. True

2. False

Q.

Find the quadratic polynomial, the sum and product of whose zeroes are -5 and 6 respectively.

Q.

Find a quadratic polynomial with the given numbers as the sum and product of its zeroes respectively.

$\sqrt{2},\frac{1}{3}$

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

1. $x^3-3x^2-10x+24$
2. $x^3+x^2+x$
3. $2x^3+x^2+1$
4. $x^3-x^2-x+2$

Q. If $\alpha$, $\beta$ are the zeros of the polynomial $f\left(x\right)=2x^2+5x+k$ satisfying the relation ${\alpha }^2+{\beta }^2+\alpha \beta =\frac{21}{4}$, then find the value of k for this condition to be true. [3 MARKS]

Q. Question 2 (v)
Are the following statements ‘True’ or False’? Justify your answer.
v) If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.

Q.

If $\alpha ,\beta$ and $\gamma$ are the zeroes of the cubic polynomial $a{x}^3+b{x}^2+cx+d$,
then $\alpha +\beta +\gamma$ is equal to

1. $\frac{c}{a}$

2. -$\frac{d}{a}$

3. $\frac{c}{d}$

4. -$\frac{b}{a}$

Q. Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and the coefficient.
$4u^2+8u$

Q. The quadratic polynomial having product of zeroes $-1$ and sum of zeroes $\frac{1}{2}$ is ____

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

Q. If one of the zero of polynomial is three more than other and sum of the zeroes is 5 then the polynomial is ____

1. $x^2+5x+4$
2. $x^2-5x-4$
3. $2x^2-5x+4$
4. $x^2-5x+4$

Q. If $\alpha ,\beta ,\gamma$ are zeros of the polynomial $a{x}^3+b{x}^2+cx+d$, then $\alpha \beta \gamma =$

1. $-\frac{c}{a}$
2. $\frac{d}{a}$
3. $-\frac{d}{a}$
4. $\frac{c}{a}$

Q. If $\alpha$ & $\beta$ are the zeroes of quad. polynomial $x^2-(k+6)x+2(2k-1)$. Find the value of $k$ so that $\alpha +\beta =\frac{1}{2}\alpha \beta$

Q. If $\alpha ,\beta$ are zeros of quadratic polynomial $k{x}^2+4x+4$, find the value of k such that $(\alpha +\beta {)}^2-2\alpha \beta =24.$

1. $k=1$ or $k=\frac{11}{3}$
2. $k=1$ or $k=\frac{1}{3}$
3. $k=-1$ or $k=\frac{2}{3}$
4. $k=-1$ or $k=\frac{10}{3}$

Q.

Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
$4,1$

Q.

If $\alpha ,\beta ,\gamma$ are the zeroes of the cubic polynomial a$x^3$+b$x^2$+cx+d = 0, then $\alpha +\beta +\gamma$ equal to:

1. 

2. -

3. -

4. 

Q. The sum$(\alpha +\beta )$ and product$\left(\alpha \beta \right)$ of zeros of polynomial $2x^2-x+5$ is _______

1. $\alpha +\beta =\frac{1}{2}\alpha \beta =\frac{5}{2}$
2. $\alpha +\beta =\frac{-1}{2}\alpha \beta =\frac{5}{2}$
3. $\alpha +\beta =\frac{1}{2}\alpha \beta =\frac{-5}{2}$
4. $\alpha +\beta =1\alpha \beta =5$

Q. The quadratic polynomial, whose zeros are $2$ and $3$, is ......

1. $x^2-5x-6$
2. $x^2+5x-6$
3. $x^2-5x+6$
4. $x^2+5x+6$

Q. Choose a polynomial having $5\&\frac{1}{2}$ as zeroes.

1. $2x^2-\frac{11}{2}x+5$
2. $2x^2-11x+2$
3. $2x^2-11x+5$
4. $2x^2-11x+\frac{5}{2}$

Q. If $\alpha$ and $\beta$ are the zeros of the polynomial $3t^2-6t+4$, find the value of $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }+2\left(\frac{1}{\alpha }+\frac{1}{\beta }\right)+3\alpha \beta$.

1. $5$
2. $8$
3. $\frac{10}{3}$
4. $\frac{1}{2}$

Q. If $\alpha$ and $\beta$ are the zeros of polynomial $x^2-2x-8$ then form a quadratic polynomial whose zeros are $3\alpha$ and 3$\beta$.

Q. Question 5
Given that one of the zeroes of the cubic polynomial $a{x}^3+b{x}^2+cx+d$ is zero, the product of the other two zeroes is
(a)$\frac{-c}{a}$
(b)$\frac{c}{a}$
(c)0
(d)$\frac{-b}{a}$

Q. If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=x^2-5x+6$, then evaluate:

$\frac{{\alpha }^2}{{\beta }^2}+\frac{{\beta }^2}{{\alpha }^2}$
[4 MARKS]

Q. If $\alpha and\beta$ are the zeroes of the quadratic polynomial $2S^2-8S+4$, then the value of $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }+2(\frac{1}{\alpha }+\frac{1}{\beta })+3\alpha \beta$ is ___.

1. 8
2. 4
3. 16
4. 12

Q. $\alpha$ and $\beta$ are the zeros of the polynomial $x^2+4x+3$. The polynomial whose zeros are $1+\frac{\beta }{\alpha }$ and $1+\frac{\alpha }{\beta }$ is

1. $x^2-16x-16$
2. $3x^2-16x-16$
3. $3x^2-16x+16$
4. $x^2-16x+16$

Q. If $\alpha \text{and}\beta$ are the zeroes of $p\left(x\right)=4x^2+10x+k$ such that ${\alpha }^2+{\beta }^2+\alpha \beta =\frac{21}{4}$, then k =

1. 12
2. 4
3. 2
4. - 12