Question

find the zeros of the quadratic polynomial and verify the relationship between the zeros and coefficients

1. x^2-2x-8​

Answer 1

$\textbf{\underline{\underline{According\:to\:the\:Question}}}$

p(x) = x² - 2x - 8

→Splitting the middle term we get :-

= x² - 4x + 2x - 8

= x(x - 4) + 2(x - 4)

⇒(x - 4)(x + 2) = 0

⇒(x - 4) = 0

⇒x = 4

⇒(x + 2) = 0

⇒x = -2

Two zeroes of the polynomial are 4 and -2

$\Large{\fbox{Relationship between zeroes and coefficient}}$

★Sum of zeroes = α + β

= 4 + (-2)

= 4 - 2

= 2

★Product of zeroes :-

= α × β

= 4 × -2

= -8

★Put,

ax² + bx + c = 0

a = 1, b = -2 and c = -8

★Sum of Zeroes :-

${\boxed{\sf\:{\alpha+\beta=-\dfrac{b}{a}}}}$

${\boxed{\sf\:{\alpha+\beta=-\dfrac{-2}{1}}}}$

= 2

★Product of zeroes :-

${\boxed{\sf\:{\alpha\times\beta=\dfrac{c}{a}}}}$

${\boxed{\sf\:{\alpha\times\beta=\dfrac{-8}{1}}}}$

= -8

$\Large{\fbox{Hence\;Verified}}$

Answer 2

Answer :

Relationship between zeroes and cofficient is verified.

Step-by-step explanation :

Let p(x) = $\sf x^2-2x-8$

Zero of the polynomial is the value of 'x'  where, p(x) = 0.

Now, Putting p(x) = 0

$\sf x^2-2x-8$

We know that :

Splitting the middle term method :

$\bigstar\;\boxed{\sf Sum =-2,Product = -8 \times 1 = -8}}$

Values in Equation :

$\sf \\ \\\implies x^2 - 4x+2x - 8 = 0\\ \\\implies x(x-4)+ 2(x-4) = 0\\ \\\implies (x+2)(x-4) = 0\\ \\So,\\ \\The\;value\;of\;x= -2,4$

$\therefore$ α = 2 & β = 4 are the zeroes of the polynomial.

⇒ (p)x = x² - 2x - 8.

⇒ 1x² - 2x - 8.

Comparing with ax² + bx +c.

So, a = 1 , b = - 2 , c = - 8.

$\rule{300}{1.5}$

★ Verification :

We know that :

$\bigstar\;\boxed{\sf Sum\;of\;zeroes = \dfrac{Cofficient\;of\;x}{Cofficient\;of\;x^2}}}$

★ L.H.S :

⇒ α + β.

⇒ - 2 + 4.

⇒ 2.

★ R.H.S :

$\sf \\ \\\implies - \dfrac{b}{a}\\ \\\implies - \dfrac{-2}{1}\\ \\\implies 2.$

$\rule{300}{1.5}$

We know that :

$\bigstar\;\boxed{\sf Product\;of\;zeroes = \dfrac{Constant\;tem}{Cofficient\;of\;x^2}}}$

★ L.H.S :

⇒ α . β.

⇒ (-2) (4).

⇒ - 8.

★ R.H.S :

$\sf \\ \\\implies \dfrac{c}{a}\\ \\\implies \dfrac{-8}{1}\\ \\\implies -8.\\ \\ \\\bigstar\;\textbf{\underline{\underline{Hence, LHS = RHS.}}}$

∴ Relationship between zeroes and cofficient is verified.