Question

find the zero of the polynomial and verify the relationship between the zeroes and the coefficients


$p^{2} -30$

Answer 1

Answer:

p²- (√30)²= 0

(p+√30) (p - √30) = 0

p+ √30 = 0 p - √30 = 0

p = -√30 p = √30

a+ b = -b/a

-√30+√30 = -0/1

0 = 0

a(b) = c/a

-√30(√30) = -30/1

-30 = -30

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Answer 2

Solution :

$\bf{\red{\underline{\bf{Given\::}}}}$

The quadratic polynomial p² - 30.

$\bf{\red{\underline{\bf{To\:find\::}}}}$

The zeroes and verify the relationship between the zeroes & coefficient.

$\bf{\red{\underline{\bf{Explanation\::}}}}$

We have p(x) = p² - 30.

Zero of the polynomial p(x) = 0

So;

$\longrightarrow\sf{p^{2} -30=0}\\\\\longrightarrow\sf{p^{2} =30}\\\\\longrightarrow\sf{p=\pm\sqrt{30} }$

∴ The α = √30 and β = -√30 are the zeroes of the polynomial.  

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

• a = 1
• b = 0
• c = -30

So;

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

$\longrightarrow\sf{\alpha +\beta =\dfrac{-b}{a} =\dfrac{Coefficient\:of\:x}{Coeeficient\:of\:x^{2} } }\\\\\\\longrightarrow\sf{\sqrt{30} +(-\sqrt{30} )=\dfrac{0}{1} }\\\\\\\longrightarrow\sf{\sqrt{30} -\sqrt{30} =0}\\\\\\\longrightarrow\sf{\pink{0=0}}$

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

$\longrightarrow\sf{\alpha \times \beta =\dfrac{c}{a} =\dfrac{Constant\:term}{Coeeficient\:of\:x^{2} } }\\\\\\\longrightarrow\sf{\sqrt{30} \times (-\sqrt{30} )=\dfrac{-30}{1} }\\\\\\\longrightarrow\sf{\sqrt{30\times (-30)}=-30}\\\\\\\longrightarrow\sf{\pink{-30=-30}}$

Thus;

Relationship between zeroes and coefficient is verified .