Question

if one of the zeroes of the cubic polynomial x3+ax2+bx+c is - 1 then find the product of other two zeroes.

Answer 1

Solution :

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Given :

One of the zeroes of the cubic polynomial x3+ax2+bx+c is - 1

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To Find :

Other zeroes of the polynomial,.

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We know that,.

The given polynomial is of the form :

ax³ + bx² + cx + d,.

If α,β & γ are the zeores of the polynomial

ax³ + bx² + cx + d, Then,.

It can also written as,

x³ + (α + β + γ)x² + (αβ + βγ + γα)x + αβγ

So,

If a = 1,

Then,

b = (α + β + γ),

c = (αβ + βγ + γα)

d = αβγ

Here,. a = 1, b = a, c = b & d = c,..

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We also know that,

Sum of the zeroes = $\frac{-b}{a}$

Let the zeroes be, α, β & -1.(as -1 is given as the zero of the given polynomial,)

⇒ α + β + (-1) = $\frac{-a}{1}$

⇒ α + β - 1 = -a

⇒ α + β = 1 - a ...(1)

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We know that,.

Product of the zeroes = $\frac{-d}{a}$

⇒ αβγ = $\frac{-c}{1}$

⇒ αβ(1) = -c

⇒ αβ = -c,.(2)

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So,

We know that,

⇒ (a + b)² = a² + 2ab + b²

Substituting zeroes of the polynomial,

We get,.

⇒ (α + β)² = α² + 2αβ + β²

⇒ (1 - a)² = α² + 2(-c) + β²

⇒ 1 - 2a + a² = α² + β² - 2c

⇒ a² - 2a + 1 -2c = α² + β² ...(3)

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Hence,

⇒ (a - b)² = a² - 2ab + b²

⇒ (α - β)² = α² - 2αβ + β²

⇒ (α - β)² = α² + β² - 2αβ

⇒ (α - β)² = a² - 2a + 1 -2c - 2(-c)

⇒ (α - β)² = a² - 2a + 1 - 2c + 2c

⇒ (α - β)² = a² - 2a + 1

⇒ (α - β)² = a² - a - a + 1

⇒ (α - β)² = a(a - 1) - 1(a - 1)

⇒ (α - β)² = (a - 1)(a - 1)

⇒ (α - β)² = (a - 1)²

⇒ ∴ α - β = a - 1...(4)

Adding Equation 1 & 4,

We get,

⇒ (α + β) + (α - β) = (1 - a) + (a - 1)

⇒ α + α + β - β = 1 - a + a - 1

⇒ 2α = 1 - 1 + a - a

⇒ 2α = 0

⇒ ∴ α = 0,.

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Substituting value of α in Equation 1,

We get,.

⇒ α + β = 1 - a

⇒ 0 + β = 1 - a

⇒ ∴ β = 1 - a


∴ The other zeroes of the given polynomial are 0 & 1 - a,.

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Sorry, the answer is lengthy & (little bit - complex),.

Answer 2

Answer:

Step-by-step explanation:

Answer:

Product of other two zeroes =c

Step-by-step explanation:

Compare p(x) with Ax³+Bx²+Cx+D , we get

A=1, B=a , C = b , D = c

Therefore,

Product of other two zeroes =c