Question

find the zeroes of polynomial and verify the relationship between zeroes and coefficients 4x2 + 8x
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Answer 1

Answer:

4x2+8x

=2x(2x+4)

=2x=0; 2x+4=0

=x=0; x=-2

let alpha be 0 and beta be -2

alpha +beta = -b/a

= 0+(-2)= -8/4

-2=-2

Alpha x beta =c/a

= 0 x(-2) = 0/4

0=0

hence , verified the relationship between zeroes and its coefficient.

Answer 2

Given:-

• A quadratic polynomial f(x) = 4x² + 8x

To Find:-

• The zeroes of the polynomial and verify the relationship between the zeroes and coefficients.

Solution:-

• We are given, f(x) = 4x² + 8x

$\sf\: \: \: \: \: \: \: \:: \implies4 {x}^{2} + 8x = 0$

$\sf \: \: \: \: \: \: \: \:: \implies4x(x + 2) = 0$

$\sf \: \: \: \: \: \: \: \:: \implies 4x = 0 \: \: \: OR\: \: \: x + 2= 0$

$\sf \pink{\: \: \: \: \: \: \: \:: \implies x = 0 \: \: \: Or\: \: \: x = - 2}$

$\therefore\:\underline{\textsf{ The zeros are \textbf{0 \: and \: -2 }}}.$

• Now, let us verify the relationship between the zeroes and coefficients.

• We know, for a quadratic polynomial f(x) = ax² + bx + c.Where:-

• Sum of zeroes = $\sf \dfrac{-b}{a}$
• Product of zeroes = $\sf \dfrac{c}{a}$

• In the given quadratic equation 4x² + 8x.

• a = 4
• b = 8
• c = 0

$\boxed{\pink{\sf Sum\ of\ the\ zeroes=\frac{-coefficient\ of\ x}{coefficient\ of\ x^{2}}}}$

$\sf Sum \: of \: zeroes = 0 + (-2) = -2$

$\sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \dfrac{-b}{a} = \dfrac{ - 8}{4} = -2$

$\boxed{\pink{\sf Product\ of\ the\ zeroes=\frac{Constant}{coefficient\ of\ x^{2}}}}$

$\sf Product \: of \: zeroes = 0 \times -2 = 0$

$\sf \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \dfrac{c}{a} = \dfrac{0}{4} = 0$

• Hence Verified..!!