Relationship between Zeroes and Coefficients of a Polynomial

Q. If $\alpha \&\beta$ are the zeroes of a quadratic polynomial p(x) and k is any constant, then what is the general form of the polynomial?

1. $k{x}^2+(\alpha +\beta )x+\alpha \beta$
2. $k{x}^2-(\alpha +\beta )x-\alpha \beta$
3. $k{x}^2+(\alpha +\beta )x-\alpha \beta$
4. $k{x}^2-(\alpha +\beta )x+\alpha \beta$

Q. If -1, 1 & 3 are the zeroes of a cubic polynomial, which of the following is the correct expression of the polynomial?

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

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

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. 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. If 𝒎 and 𝒏 are zeroes of the polynomial 𝒇(𝒙)=𝟐𝒙²+𝟔𝒙+𝒌, find the value of 𝒌, such that $m^2+n^2=29$.

1. -20
2. -10
3. 10
4. 20

Q. If p, q and r are the roots of a cubic polynomial $a{x}^3+b{x}^2+cx+d$, then what will be pqr?

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

Q. If p, q & r are the zeroes of a cubic polynomial $a{x}^3+b{x}^2+cx+d$, then what will be p+q+r?

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

Q. If $\alpha \&\beta$ are the zeroes of a quadratic polynomial p(x), and k is any constant, then what is the general form of the polynomial?

1. $k\left[x^2-(\alpha +\beta )x+\alpha \beta \right]$
2. $k{x}^2-(\alpha +\beta )x-\alpha \beta$
3. $k{x}^2+(\alpha +\beta )x-\alpha \beta$
4. $k^2{x}^2+k(\alpha +\beta )x+k\alpha \beta$

Q. If p, q & r are the roots of a cubic polynomial $a{x}^3+b{x}^2+cx+d$, then what will be p+q+r?

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