Relationship Between Zeroes and Coefficients of a Cubic Polynomial

Q. Obtain all other zeroes of $3x^4+6x^3-2x^2-10x-5$, if two of its zeroes are $\sqrt{\frac{5}{3}}and-\sqrt{\frac{5}{3}}$.

Q.

Find the polynomial whose sum of its zeroes is $-\frac{8}{5}$ and the products of the zeroes is $\frac{7}{5}$

1. 

2. 

3. 

4. 

Q.

Write the polynomial whose zeroes are $-2-\sqrt{3}\text{and}2-\sqrt{3}$.

Q. Question 2 (vi)
Are the following statements ‘True’ or False’? Justify your answer.
vi) If all three zeroes of a cubic polynomial $x^3+a{x}^2–bx+c$ are positive, then atleast one of a, b and c is non - negative.

Q.

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

$t^2-15$

Q. If $\alpha ,\beta ,\gamma$ are the zeroes of the cubic polynomial $x^3+4x+2$, then find the value of:
$\frac{1}{\alpha +\beta }+\frac{1}{\beta +\gamma }+\frac{1}{\gamma +\alpha }$

Q. The polynomial with $1,\frac{1}{2}\&\frac{-1}{2}$ as zeroes is ______

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

Q.

If the sum of zeros of the polynomial p(x) = $k{x}^3-5x^2-11x-3$ is 2, then k is equal to:

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

2. $\frac{2}{5}$

3. 3

4. $\frac{5}{2}$

Q. Find the remaining two zeros of the polynomial $x^4-x^3-13x^2+7x+42$ if two of its zero are $\sqrt{7},-\sqrt{7}$.

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

Q. Question 2 (vi)
Are the following statements ‘True’ or False’? Justify your answer.
vi) If all three zeroes of a cubic polynomial $x^3+a{x}^2–bx+c$ are positive, then atleast one of a, b and c is non - negative.

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. Verify that $1,-1$ and $-3$ are the zeroes of the cubic polynomial $x^3$ $+3x^2-x-3$ and check the relationship between zeroes and the coefficient.

Q. State 'T' for true and 'F' for false.
I. If the square of the sum of the zeroes of a quadratic polynomial is 4 times the product of them, then the zeroes are equal in magnitude.
II. A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 4, then the linear term of the polynomial doesn't exist.
III. If the sum of the zeroes of a cubic polynomial is 5 and the sum of the product of two zeroes is 8, then the sum of the square of the zeroes will be 10.
IV. If one of the roots of the polynomial f(x) = $6x^2+13x+k$ is reciprocal of the other, then k is 6.

1. I - T II - F III - T IV - F
2. I - F II - T III - F IV - T
3. I - T II - F III - F IV - T
4. I - T II - T III - F IV - T

Q. Question 1 (ii)
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.

iii) $-2\sqrt{3},-9$

Q.

Find a cubic polynomial with sum of its zeroes, sum of the product of its zeroes taken two at a time and product of its zeroes as $\frac{5}{3},\frac{-11}{3}$ and $1$ respectively.

Q. If one of the zeros of the cubic polynomial $x^3+a{x}^2+bx+c$ is $-1$, then the product of the other two zeros is

1. $b-a+1$
2. $a-b+1$
3. $b-a-1$
4. $a-b-1$

Q. The quadratic polynomial whose sum and product of the zeroes is 9 and $\frac{1}{9}$ respectively is .

1. x2-9x+19
2. x2+9x-19
3. -x2-9x+19
4. 9x2-9x+19

Q.

How do you find a polynomial of degree $3$ that has zeros of $-3$, $0$, $1$?

Q.

If k and 2k are zeros of $f\left(x\right)=x^3+4x^2+9kx=90$, find k and all three zeros of f(x).

1. K = 1, roots = -3, -6, -5

2. K = -3, roots = -3, -6, 5

3. K = -3, roots = -3, 6, -5

4. K = 3, roots = -3, -6, 5

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

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

Q.

Write the sum and product of the zeroes of the polynomial $p\left(x\right)=x^3-4x$.

Q. Question 1 (iv)
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.
iv) $=\frac{-3}{2\sqrt{5}},-\frac{1}{2}$

Q. If the sum and the product of zeros of a quadratic polynomial $k{x}^2-(k+1)x+(t-1)$ is 5 and 6, respectively, find the value of k $\times$ t.

1. 0
2. $-\frac{1}{8}$
3. $-\frac{1}{3}$
4. $\frac{5}{8}$

Q. The sum and the product, respectively, of the zeros of the quadratic polynomial $x^2+8x+16$ are ____ and ____.

1. 16, 8
2. 8, 16
3. -8, 16
4. -8, -16

Q.

Form the polynomial whose zeroes are $\frac{6+\sqrt{3}}{3},\frac{6-\sqrt{3}}{3}$

1. 

2. 

3. 

4. 

Q. α = 1, β = -1 and γ = 0 then αβ + βγ + γα =

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

Q. If p, q & r are the roots of a cubic polynomial
$a{x}^3+b{x}^2+cx+d$, then pq+qr+rp = $\frac{d}{a}$.

1. True
2. False

Q. Question 2 (vi)
Are the following statements ‘True’ or False’? Justify your answer.
vi) If all three zeroes of a cubic polynomial x3 + ax2 –bx + c are positive, then atleast one of a, b and c is non - negative.

Q. If the sum and the product of zeros of a quadratic polynomial $k{x}^2-(k+1)x+(t-1)$ is 5 and 6, respectively, find the value of k $\times$ t.

1. 0
2. $-\frac{1}{8}$
3. $-\frac{1}{3}$
4. $\frac{5}{8}$

Q. If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=6x^2-x-7$, then evaluate: [3 MARKS]

(i) ${\alpha }^2+{\beta }^2$

(ii) $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$