Question

verify that 2,1,1are the zeroes of cubic polynomial p(x)=x^3-4x^2+5x-2 and verify the relationship between the zeroes and coefficients
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Answer 1

Answer:

Refers to the attachment....

Answer 2

Answer:-

• Given:-

Zeroes of a cubic polynomial are = 2, 1, 1

Cubic polynomial = p(x) = x³ - 4x² + 5x - 2

• Solution:-

i] Comparing the given polynomial with ax³ + bx² + cx + d ,

we get,

a = 1

b = -4

c = 5

d = -2

Further,

Given that,

2 , 1 ,1 are zeroes of the given polynomial

HENCE

p(2) = 1 × (2)³ + (-4) (2)² + 5 (2) + (-2)

→ 8 + (-4 × 4) + 10 - 2

→ 8 - 16 + 10 - 2

→ 18 - 18

→ 0

Now,

p(1) = (1)³ + (-4) (1)² + 5(1) + (-2)

→ 1 + (-4 × 1) + 5 - 2

→ 1 - 4 + 5 - 2

→ 6 - 6

→ 0

Therefore, [2 , 1 ,1 ] are the zeroes of the given polynomial.

ii] So,

We take $\bf{\alpha = 2 , \beta = 1 , \gamma = 1}$

Now,

$\alpha + \beta + \gamma$ = 2 + 1 + 1

→ 5 = $\bf{\dfrac{-b}{a}}$

$\alpha \beta + \beta \gamma + \gamma \alpha$ = 2 × 1 + 1 × 1 + 1 × 2

→ 2 + 1 + 2

→ 5 = $\bf{\dfrac{c}{a}}$

$\alpha\beta\gamma$ = 2 × 1 × 1

→ 2 = $\bf{\dfrac{-d}{a}}$

HENCE VERIFIED