Question

find the zeroes of the polynomial 3x+2x-5 and verify the relationship between the zeroes and coefficients

Answer 1

S O L U T I O N :

We have p(x) = 3x² + 2x - 5

Zero of the polynomial p(x) = 0

So;

$\longrightarrow\rm{3x^{2} +2x-5=0}\\\\\longrightarrow\rm{3x^{2} -3x+5x-5=0}\\\\\longrightarrow\rm{3x(x-1)+5(x-1)=0}\\\\\longrightarrow\rm{(x-1)(3x+5)=0}\\\\\longrightarrow\rm{x-1=0\:\:\:Or\:\:\:3x+5=0}\\\\\longrightarrow\rm{x=1\:\:\:Or\:\:\:3x=-5}\\\\\longrightarrow\bf{x=1\:\:\:Or\:\:\:x=-5/3}$

∴ The α = 1 and β = -5/3 are the zeroes of the polynomial.

As the given quadratic polynomial as we compared with ax² + bx + c;

• a = 3
• b = 2
• c = -5

Now;

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

$\mapsto\sf{\alpha +\beta =\dfrac{-b}{a} =\dfrac{Coefficient\:of\:x}{Coefficient\:of\:x^{2} } }\\\\\\\mapsto\sf{1+\bigg(-\dfrac{5}{3} \bigg)=\dfrac{-2}{3}} \\\\\\\mapsto\sf{1-\dfrac{5}{3} =\dfrac{-2}{3} }\\\\\\\mapsto\sf{\dfrac{3-5}{3} =\dfrac{-2}{3} }\\\\\\\mapsto\bf{\dfrac{-2}{3} =\dfrac{-2}{3} }$

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

$\mapsto\sf{\alpha \times \beta =\dfrac{c}{a} =\dfrac{Constant\:term}{Coefficient\:of\:x^{2} } }\\\\\\\mapsto\sf{1\times \bigg(-\dfrac{5}{3} \bigg)=\dfrac{-5}{3}} \\\\\\\mapsto\bf{\dfrac{-5}{3} =\dfrac{-5}{3} }$

Thus;

Relationship between zeroes and coefficient is verified .