Question

find the zeros of the polynomials and verify the relationship between zeroes and its coefficient 3xsquare +4x-4​

Answer 1

$\star\;\;{\underline{\underline{\frak \pink{GiVeN:-}}}}$

• p(x) = 3x² + 4x + 4

$\star\;\;{\underline{\underline{\frak \pink{To\;Find:-}}}}$

• Zeroes of polynomial and to verify relationship b/w zeroes and coefficient.

$\star\;\;{\underline{\underline{\frak \red{SolutiOn:-}}}}$

$\normalsize{\underline{\underline{\sf{\dag\;Using\;middle\;term\; splitting\;method}}}}$

$:\implies$ 3x² + 4x - 4

$:\implies$ 3x² + 6x - 2x - 4

$:\implies$ 3x(x + 2) - 2(x + 2)

$:\implies$ (3x - 2)(x + 2)

Therefore, zeroes are:-

$:\implies$ (3x - 2) = 0

${\boxed{\sf{\red{\leadsto \; x = \dfrac{2}{3}}}}}$

And

$:\implies$ (x + 2) = 0

${\boxed{\sf{\red{\leadsto \; x = - 2}}}}$

Let $\sf \alpha = \dfrac{2}{3}$

Let $\sf \beta = - 2$

$\rule{200}3$

$\dag\;\;\normalsize{\underline{\underline{\sf{\purple{Verification:-}}}}}$

★ Relationship b/w zeroes and coefficient:-

We know that,

✠ Sum of zereos ( $\alpha + \beta$ ) = $\dfrac{-b}{a}$

$\dashrightarrow\sf -2 + \dfrac{2}{3} = \dfrac{-4}{3}$

$\dashrightarrow\sf \underbrace{ \dfrac{-4}{3} = \dfrac{-4}{3}}_{LHS = RHS}$

✠ Product of zereos ( $\alpha \beta$ ) = $\dfrac{c}{a}$

$\dashrightarrow\sf -2 \times \dfrac{2}{3} = \dfrac{-4}{3}$

$\dashrightarrow\sf \underbrace{ \dfrac{-4}{3} = \dfrac{-4}{3}}_{LHS = RHS}$

$\dag\;\;\normalsize{\underline{\underline{\sf{\pink{Hence\; Verified!!}}}}}$

$\rule{200}3$