Relationship between Zeroes and Coefficients of a Polynomial

Q.

If one zero of polynomial (a^{​​​​​​2}​​​​​+9)x^{​​​​​​2}+13x+6a is reciprocal of other find value of 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 $2,-7,-14$ respectively.

Q.

Given that the zeroes of the cubic polynomial $x^3-6x^2+3x+10$ are of the form$a,a+b,a+2b$ for some real numbers $a$ and $b$, find the values of $a$ and $b$ as well as the zeroes of the given polynomial.

Q.

If one zero of the polynomial $p\left(x\right)=\left(a^2+9\right)x^2+45x+6a$ is reciprocal of the other, find the value of $a$.

Q.

Find a quadratic polynomial with $\frac{1}{8}$ as the sum and 2 as product of its zeroes.

1. $8x^2-x-16$

2. $8x^2+x+16$

3. $8x^2+x-16$

4. $8x^2-x+16$

Q. Find the sum of the zeroes of the following polynomial:
P($x$) = $4x^2$ - 1

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

Q. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as $2,7,14$ respectively.

Q. The zeros of the quadratic polynomial $x^2+99x+127$ are

1. Both positive
2. Both negative
3. One positive and one negative
4. Both equal

Q. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
(i) $2x^3+x^2-5x+2;\frac{1}{2},1,-2$

Q.

Write a polynomial $p\left(x\right)$, whose sum and product of the zeroes are $\frac{9}{4}$and $\frac{5}{8}$respectively

Q. Find the zeroes of the following polynomial by factorisation method and verify the relation between the zeroes and the coefficients of the polynomials:

$2x^2+\frac{7}{2}x+\frac{3}{4}$.

1. $\frac{-1}{2},\frac{-1}{4}$
2. $\frac{-3}{2},\frac{-1}{4}$
3. $\frac{-7}{2},\frac{-1}{4}$
4. $\frac{-9}{2},\frac{-1}{4}$

Q. If $x=-\frac{1}{3}$ is a zero of a polynomial $p\left(x\right)=27x^3-{ax}^2-x+3$, then find the value of ${}^{\prime }{a}^{\prime }$

Q. If $\alpha$ and $\beta$ are the zeroes of the polynomial $f\left(x\right)=x^2-6x+k$, find the value of $k$, such that ${\alpha }^2+{\beta }^2=40$

Q. If the zeroes of the quadratic polynomial $x^2+(a+1)x+b$ are $2$ and $3$, then

1. $a=2$, $b=6$
2. $a=7$, $b=1$
3. $a=5$, $b=1$
4. $a=-6$, $b=6$

Q. The polynomial whose zeroes are $\frac{5+\sqrt{3}}{3},\frac{5-\sqrt{3}}{3}$ is .

1. 9x2-30x+22
2. 9x2+30x+22
3. x2+30x+22

Q.

1. 
   
   $\frac{7}{2},\frac{-1}{6}$
2. 
   
   $\frac{5}{2},\frac{1}{8}$
3. 
   
   $\frac{3}{2},\frac{3}{7}$
4. 
   
   $\frac{3}{4},\frac{2}{9}$

Q.

Given that two of the zeroes of the cubic polynomial $a{x}^3+b{x}^2+cx+d$are $0$, then find the third zero?

Q.

The sum of all zeroes of the polynomial p(x) = $5x^5-20x^4+5x^3+50x^2-20x-40$ is

___

and the product is

___ .

Q.

‘- p’ and ‘q’ are the zeroes of the polynomial $x^2–bx+c$. The zeroes of the polynomial $x^2–bxy+c{y}^2$are:

1. py, -qy

2. -py, qy

3. -py, qy

4. py, qy

Q. If $2$ and $3$ are zeros of polynomial $3x^2-2kx+2m$, then what is the value of $m$?

Q. Find the zeroes of the following quadratic polynomial and verify.
$x^2-2x-15$

Q. Find the zeros of the quadratic polynomial $f\left(x\right)=x^2-3x-28$ and verify the relationships between the zeros and the coefficients.

1. $x=7,-4$
   
2. product of roots $=-28$
3. sum of roots $=-3$
4. $x=-7,4$

Q. Find the zero of the polynomial in each of the following.

a. $p\left(x\right)=3x-2$
b. $3x$
c. $ax,a\ne 0$
d. $cx+d,c\ne 0,c,d$ are real numbers.

Q. Sum of the zeros of the cubic polynomial
P(x)=$3x^3-5x^2-11x-3$ is:

1. $3$
2. 0
3. $11$
4. $\frac{5}{3}$

Q. Find the quadratic polynomial whose zeros are given as following
$\frac{1}{4},-1$

Q.
If the zeros of the polynomial $f\left(x\right)=x^3-3x^2+x+1$ are a-b, a, a+b, find a and b.

Q.

Q. The zeroes of the polynomial $6x^2+17x-88$ are

1. $-\frac{8}{3},\frac{11}{2}$
2. $-\frac{11}{2},\frac{8}{3}$
3. $2,-\frac{9}{2}$
4. $-2,\frac{9}{2}$

Q. Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and their coefficients.

1. $-\frac{1}{3},\frac{1}{2}$
2. $-\frac{1}{4},\frac{1}{2}$
3. $-\frac{1}{9},\frac{1}{7}$
4. $-\frac{1}{3},\frac{1}{3}$

Q. If one zero of the polynomial $x^2-4x+1$ is $(2+\sqrt{3})$ write the other zero.