Question

Show that 1/2 and -3/2 are the zeros of polynomial for 4x square +4x -3 and verify relationship between zeros and coefficient

Answer 1

$given \: polynomial \: is \: p(x) = 4x {}^{2} + 4x - 3 \\ \\ = 4x {}^{2} + 6x - 2x - 3 \\ \\ = 2x(2x + 3) - 1(2x + 3) \\ \\ = (2x + 3)(2x - 1) \\ \\ to \: find \: the \: zeroes \: of \: the \: given \: polynomial \\ \\ p(x) = 0 \\ \\ therefore \: \: (2x + 3)(2x - 1) = 0 \\ \\ therefore \: \: (2x + 3) = 0 \: \: and \: \: (2x - 1) = 0 \\ \\ x = \frac{ - 3}{2} \: \: and \: x = \frac{1}{2} \\ \\ hence \: \: \frac{1}{2} \: and \: \frac{ - 3}{2} are \: the \: zeroes \: of \: the \: given \: polynomial. \\ \\ \\ \: \: proved \\ \\ hope \: it \: helps...$

Answer 2

to check whether the given zeroes are the zeroes of this polynomial or not we will put values in the equation :
p(1/2)=4x^2+4x-3=0
p(1/2)=4(1/2)^2+4(1/2)-3=0
p(1/2)=3-3=0

p(-3/2)=4(-3/2)^2+4(-3/2)-3
p(-3/2)=9-6-3=0

Hence, 1/2 as well as -3/2 are zeroes of the given polynomial.
Verification of relationship between zeroes and coefficients:
sum of zeroes= 1/2-3/2=1-3/2=-1
=-b/a
=-4/4=-1
-1=-1
product of zeroes: 1/2(-3/2)=-3/4
=c/a
=-3/4
-3/4=-3/4
Hence, proved.