Question

Show that 1 /2 and −3/ 2 are the zeroes of the polynomial 42+4x-3 and verify the relationship between the zeroes and coefficients of polynomial

Answer 1

Step-by-step explanation:

Hope it helps you to clarify your doubt

Answer 2

Solution :

$\bigstar$We have quadratic polynomial p(x) = 4x² + 4x - 3

Zero of the polynomial p(x) = 0

Show :

$\longrightarrow\sf{4x^{2} +4x-3=0}\\\\\longrightarrow\sf{4x^{2} -2x+6x-3=0}\\\\\longrightarrow\sf{2x(2x-1)+3(2x-1)=0}\\\\\longrightarrow\sf{(2x-1)(2x+3)=0}\\\\\longrightarrow\sf{2x-1=0\:\:\:Or\:\:\:2x+3=0}\\\\\longrightarrow\sf{2x=1\:\:\:Or\:\:\:2x=-3}\\\\\longrightarrow\bf{x=1/2\:\:\:Or\:\:\:x=-3/2}$

∴ α = 1/2 & β = -3/2 are the zeroes of the polynomial.

As we know that given polynomial compared with ax² + bx + c;

• a = 4
• b = 4
• c = -3

Now;

$\underline{\mathcal{SUM\:OF\:THE\:ZEROES\::}}$

$\mapsto\tt{\alpha +\beta =\dfrac{-b}{a} =\bigg(\dfrac{Coefficient\:of\:x}{Coefficient\:of\:x^{2} } \bigg)}\\\\\\\mapsto\tt{\dfrac{1}{2} +\bigg(-\dfrac{3}{2} \bigg)=\cancel{\dfrac{-4}{4}} }\\\\\\\mapsto\tt{\dfrac{1}{2} -\dfrac{3}{2} =1}\\\\\\\mapsto\tt{\dfrac{1-3}{2} =-1}\\\\\\\mapsto\sf{\cancel{\dfrac{-2}{2}} =-1}\\\\\mapsto\bf{-1=-1}$

$\underline{\mathcal{PRODUCT\:OF\:THE\:ZEROES\::}}$

$\mapsto\tt{\alpha \times \beta =\dfrac{-b}{a} =\bigg(\dfrac{Constant\:term}{Coefficient\:of\:x^{2} } \bigg)}\\\\\\\mapsto\tt{\dfrac{1}{2} \times \bigg(-\dfrac{3}{2} \bigg)=\dfrac{-3}{4} }\\\\\\\mapsto\bf{-\dfrac{3}{4} =\dfrac{-3}{4}}$

Thus;

The zeroes & coefficient of zeroes & relationship are verified .