Question

Given that x − √5 is the factor of the polynomial x^3 − 3√5x^2 − 5x + 15√5 .find all the zeroes of the polynomial.

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Answer 1

Step-by-step explanation:

Given -

• x - √5 is the factor of the polynomial p(x) = x³ - 3√5x² - 5x + 15√5

To Find -

• Zeroes of the polynomial

Now,

x-√5)x³-3√5x²-5x+15√5(x²-2√5x-15

x³ - √5x²

(-) (+)

_____________________

-2√5x² - 5x + 15√5

-2√5x² + 10x

(+) (-)

_________________

-15x + 15√5

-15x + 15√5

(+) (-)

__________

X X

Now, Factorising x² - 2√5x - 15, we get :

→ x² - 2√5x - 15

here,

a = 1

b = -2√5

c = -15

By using quadratic formula :-

• x = -b ± √b² - 4ac/2a

→ -(-2√5) ± √(-2√5)² - 4×1×-15/2(1)

→ 2√5 ± √20 + 60/2

→ 2√5 ± √80/2

→ 2√5 ± 4√5/2

Zeroes are -

→ x = 2√5 + 4√5/2

→ 6√5/2

→ 3√5

And

→ x = 2√5 - 4√5/2

→ -2√5/2

→ -√5

Hence,

The zeroes of polynomial p(x) = x³ - 3√5x² - 5x + 15√5 is 3√5, √5, -√5.

Verification :-

• α + β + γ = -b/a

→ 3√5 + √5 - √5 = -(-3√5)/1

→ 3√5 = 3√5

LHS = RHS

And

• αβ + βγ + γα = c/a

→ 3√5×√5 + √5×-√5 + -√5×3√5 = -5/1

→ 15 - 5 - 15 = -5

→ -5 = -5

LHS = RHS

And

• αβγ = -d/a

→ 3√5 × √5 × -√5 = -(15√5)/1

→ -15√5 = -15√5

LHS = RHS

Hence,

Verified..

It shows that our answer is absolutely correct.

Answer 2

Step-by-step explanation:

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