Relationship Between Zeros And Coefficients Of A Polynomial

Relationship between Zeros and coefficients of a Polynomial

zeros of a polynomial : A real number $\alpha$ is called a zero of the polynomial $p(x)$

, if $p(\alpha )=0$ .

If “ $\alpha$ ” is a zero of a polynomial $p(x)$ , then by factor theorem $(x-\alpha )$ is a factor of a given polynomial. The relation between the zeros and the coefficients of a polynomial is given below.

Linear Polynomial:

The linear polynomial is an expression , in which the degree of the polynomial is 1 . The general form of a linear polynomial is $ax+b$ . Here, $x$ is a variable, “a” and “b” are constant.

Let $\large p(x)=ax+b, \, a\neq 0$ be a linear polynomial,

then means .

$\Rightarrow x=-\frac{b}{a}$

So $x=-\frac{b}{a}$ = $-\frac{constant \, \, \, term}{coefficient \, \, of\, \, x}$ .

Quadratic polynomial:A polynomial of degree 2 is called a quadratic polynomial. A quadratic polynomial in one variable will have at most tree terms. Any quadratic polynomial in $x$ will be of the form $\large ax^{2}+bx+c\, \, where \, \, a\neq 0\, \, and \, \, a,b,c\, \, are\, \, constants.$

Let $\alpha$ and $\beta$ be the zeros of the quadratic polynomial $p(x)=ax^{2}+bx+c, \, \, a\neq 0$ .

Then $(x-\alpha )$ and $(x-\beta )$ are factors of $p(x)$ . $\therefore \, \, \, ax^{2}+bx+c=k(x-\alpha)(x-\beta )$ , where $k$ is a constant.

$\, \, \, \, \, \, \, =k\left [ x^{2}-(\alpha +\beta )x+\alpha \beta \right ]$

On comparing coefficients of like powers of $x$ on both sides, we get

$k=a, \, \, -k(\alpha +\beta )=b,\, \, k(\alpha \beta) =c$

$\Rightarrow -a(\alpha +\beta )=b \, \, and \, \, \, a(\alpha \beta )=c$ $(\because k=a)$

$\Rightarrow \alpha +\beta =-\frac{b}{a} \, \, \, and\, \ \alpha\beta =\frac{c}{a}$

$\therefore$ sum of zeros = – $\frac{(cofficient\, \,of\, \, x)}{(coefficient \, \, of \, \, x^{2})},$

product of zeros = $\frac{constant\, \,term }{coefficient\, \,of \, \,x^{2}}$

Cubic polynomial : A polynomial of degree 3 is called cubic polynomials. Any cubic polynomial can have at most 4 terms. Cubic polynomial can be written in the form $ax^{3}+bx^{2}+cx+d, a\neq 0$ and $a,\, \, b,\, \, c$ and $d$ are constants.

Let $\alpha$ , $\beta$ and $\gamma$ be the zeros of the cubic polynomial $p(x)=ax^{3}+bx^{2}+cx+d, \, \, a\neq 0$ .

Then $(x-\alpha )$ , $(x-\beta )$ and $(x-\gamma )$ are factors of $p(x)$ .

$\therefore \, \, \, ax^{3}+bx^{2}+cx+d=k(x-\alpha)(x-\beta )(x-\gamma )$ , for some constant $k$ .

$\, \, \, \, \, \, \, =k\left [ x^{3}-(\alpha +\beta+\gamma )x^{2}+(\alpha \beta +\beta \gamma +\alpha \gamma )x-\alpha \beta \gamma \right ]$

= $kx^{3}-k(\alpha +\beta +\gamma )x^{2}+k(\alpha \beta +\beta \gamma +\gamma \alpha )x-k\alpha \beta \gamma$

On comparing coefficients of like powers of $x$ on both sides, we get

$k=a, \, \, -k(\alpha +\beta+\gamma )=b,\, \, k(\alpha \beta+\gamma +\gamma \alpha ) =c,\, \, -k(\alpha \beta \gamma )=d$

$\Rightarrow -a(\alpha +\beta+\gamma )=b, \, \, a(\alpha \beta+\beta \gamma +\gamma \alpha )=c,\, \, -a(\alpha \beta \gamma ) =d$ $(\because k=a)$

$\Rightarrow \alpha +\beta +\gamma =-\frac{b}{a}, \, \, \alpha\beta+\beta \gamma+\gamma \alpha =\frac{c}{a}, \, \,\alpha \beta \gamma =-\frac{d}{a}$

$\therefore$ If $\alpha$ , $\beta$ and $\gamma$ be the zeros of the cubic polynomial $p(x)=ax^{3}+bx^{2}+cx+d, \, \, a\neq 0$ , then

(i) $\alpha +\beta +\gamma =\frac{b}{a}$ (ii) $\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a}$ (iii) $\alpha \beta \gamma =-\frac{d}{a}$

Similarly, If α , β, γ, δ are roots of the equation $ax^{4}+ bx^{3} + cx^{2} + dx +e=0, a\neq 0$ , then

$\alpha +\beta +\gamma +\delta =-\frac{b}{a}$

$\alpha \beta + \beta \gamma+\gamma \delta +\delta \alpha +\delta \beta +\gamma \alpha =\frac{c}{a}$

$\alpha \beta \gamma +\alpha \gamma \delta +\alpha \beta \delta +\beta \gamma \delta=-\frac{d}{a}$

$\alpha \beta \gamma \delta =\frac{e}{a}$ .

Some practice questions based on polynomial are given in the following worksheet.

download link: polynomial worksheet

Relationship between Zeros and coefficients of a Polynomial

Bina singh

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Bina singh

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