Question

Compute the zeroes of the polynomial 4x 2 – 4x – 8. Also, establish a relationship between the
zeroes and coefficients.

Answer 1

4x 2 – 4x – 8

4x^2-8x+4x-8

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

4x-4=0. x-2=0

4x=4. x=2

x=4/4. x=2

x=1. x=2

1,2are the zeroes of the polynomial 4x 2 – 4x – 8

Relationship between zeroes and coefficients

Sum of zeroes=-(coefficient of x)/coefficient of x^2=-(-4)/4=4/4=1

Product of zeroes=constant term/coefficient of x^2=-8/4=-2

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Answer 2

$\underline\mathfrak{Given:-}$

• The zeroes of the polynomial 4x² – 4x – 8.

$\underline\mathfrak{To \: \: Find:-}$

• Find the zeroes and co-efficent ...?

$\underline\mathfrak{Solutions:-}$

$\: \: \: \: \: \: \: \leadsto \: \: {4x}^{2} \: - \: {4x} \: - \: {8} \: \: = \: \: {0}$

$\: \: \: \: \: \: \: \leadsto \: \: {4x}^{2} \: - \: {8x} \: + \: {4x} \: - \: {8} \: \: = \: \: {0}$

$\: \: \: \: \: \: \: \leadsto \: \: {4x} \: {({x} \: - \: {2})} \: + \: {4x} \: {({x} \: - \: {2})} \: \: = \: \: {0}$

$\: \: \: \: \: \: \: \leadsto \: \: {({4x} \: + \: {4})} \: \: \: {({x} \: - \: {2})} \: \: = \: \: {0}$

$\: \: \: \: \: \: \: \leadsto \: \: {x} \: \: = \: \: {-1} \: \: \: {x} \: \: = \: \: {2}$

$\: \: \: \: \: \: \: \leadsto \: \: {\alpha} \: \: = \: \: {-1} \: \: \: {\beta} \: \: = \: \: {2}$

Hence, the value of zeroes is -1 and 2.

The equation is in the form of:-

⟹ 4x² - 4x - 8 = 0

• a = 4
• b = -4
• c = -8

$\: \: \: \: \: \therefore {Sum \: \: of \: \: zeroes} \: \: = \: \: \frac{ \: - \: coefficient \: \: of \: \: x}{coefficient \: \: of \: \: {x}^{2}}$

$\: \: \: \: \: \leadsto \: \: \alpha \: \: + \: \beta \: \: = \: \: \frac{-b}{a}$

$\: \: \: \: \: \leadsto \: \: {-1} \: \: + \: {2} \: \: = \: \: - \: \frac{-4}{4}$

$\: \: \: \: \: \leadsto \: \: {1} \: \: = \: \: {1}$

$\: \: \: \: \: \therefore {Product \: \: of \: \: zeroes} \: \: = \: \: \frac{constant \: \: term}{coefficient \: \: of \: \: {x}^{2}}$

$\: \: \: \: \: \leadsto \: \: \alpha\beta \: \: = \: \: \frac{c}{a}$

$\: \: \: \: \: \leadsto \: \: {-1} \: \times \: {2} \: \: = \: \: \frac{-8}{4}$

$\: \: \: \: \: \leadsto \: \: {-2} \: \: = \: \: {-2}$

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