Question

Explain in detail about the relationship between zeroes & coefficients of a polynomial.

Points : 15 ☺

Answer 1

polynomial :-
polynomial is in the form of
f(x) = aoxⁿ + a1.x^(n-1) +a2.x^(n-2)........+an.x^0

where ao, a1, a2, a3 , a4, .......an are real numbers . where ao ≠ 0
and , power of x is not be fractional , square root ,
because in this ways polynomial , in some undefined .
e.g √x x is not possible here negative.

degree of polynomial :-
highest power of x in polynomial is known as degree of polynomial .

example :- 2x³ +5x² -6x +7 =0
here we see that highest power of x is 3
so, degree of polynomial is 3 .

coefficient of polynomial :-
in polynomial all x attached with constant . this constant is known as coefficient of polynomial .
★ no of degree of polynomial = no of zero of polynomial

example :- 2x⁴ + 4x +3

here coefficient of x⁴ = 2
coefficient of x = 4

relationship between zeros and coefficient of polynomial .
=============================
in 10th syllabus you have to read two or 3 degree polynomial so, I explained only this .

for two dgree polynomial
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this is also known as quadratic polynomial .

quadratic polynomial standard form just like -----

ax² + bx +c
here a , b are the coefficient of x² and x respectively. and c is the constant .

because quadratic are two degree polynomial so, no of zeros equal =2

let A and B is the zeros of quadratic polynomial .
then ,

sum of zeros = -coefficient of x/coefficient of x²

A + B = -b/a

product of zeros = constant/coefficient of x²

A.B = c/a


for 3 degree polynomial
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3 degree polynomial standard form is

ax³ + bx³ + cx + d

3 degree polynomial have 3 zeros . let A , B , and C are zeros .

then,
sum of zeros = -coefficient of x²/coefficient of x³

A + B + C = -b/a

sum of products of two zeros = coefficient of x/coefficient of x³

AB + BC +CA = c/a

product of zeros = - constant/coefficient of x³

ABC = -d/a

Answer 2

RELATION BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL:

1) the zero of a polynomial .ax+b is -b/a.
i.e x = - (constant term )/ coeffiecient of x

ex : if p(x)= 5x +7 
zero of p(x) = -7/5

2) let α, β are two zeroes of a quadratic polynomial, ax²+bx+c, (a≠0) then
i) sum of the zeroes = α+β = -b/a = - (x - coefficient)/(x² coefficient)
ii) product of the zeroes = αβ = c/a = constant / (x² coefficient)

3) Let α,β, and gamma are the zeroes of cubic polynomial ax³+bx²+cx+d, (a≠0), then

i) α+β+gamma = -b/a = - (x² - coefficient )/ (x³ coefficient)
ii) αβ+β*gamma + gamma * α = c/a = (x - coefficient ) / (x³ - coefficient)
iii) αβ*gamma = -d/a = - (constant term) /( co efficient of x³)