December 8, 2022

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 “bare constant.

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

thenp(x)=0   means    ax+b=0.

                                                \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

 

 

 

 

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