Question

verify that 5,-2 and 1/3 are the zeros of the cubic polynomial p(x)=3x^3-10x^2 -27x +10 and verify the relation between its zeroes and coefdicients .​

Answer 1

Answer :-

We have :-

→ p(x) = 3x³ - 10x² - 27x + 10

→ It's zeroes are 5, -2 and 1/3 .

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5, -2 and 1/3 will be the zeroes of p(x) if p(5) = 0 ; p(-2) = 0 ; p(1/3) = 0.

⇒ p(5) = 3(5)³ - 10(5)² - 27(5) + 10

⇒ p(5) = 375 - 250 - 135 + 10

⇒ p(5) = 0

⇒ p(-2) = 3(-2)³ - 10(-2)² - 27(-2) + 10

⇒ p(-2) = -24 - 40 + 54 + 10

⇒ p(-2) = 0

⇒ p(1/3) = 3(1/3)³ - 10(1/3)² - 27(1/3) + 10

⇒ p(1/3) = 1/9 - 10/9 - 9 + 10

⇒ p(1/3) = 0

So, 5, -2 and 1/3 are the zeroes of p(x) .

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If we compare p(x) with ax³ + bx² + cx + d , we get :-

→ a = 3 ; b = -10 ; c = -27 ; d = 10

Let , α = 5 ; β = -2 ; γ = 1/3 .

⇒ (α + β + γ) = [5 + (-2) + 1/3]

⇒ (α + β + γ) = [(15 - 6 + 1)/3]

⇒ (α + β + γ) = 10/3

⇒ (α + β + γ) = -b/a

⇒ (αβ + βγ + γα) = [5(-2) + (-2)(1/3) + 1/3(5)]

⇒ (αβ + βγ + γα) = [-10 - 2/3 + 5/3]

⇒ (αβ + βγ + γα) = [(-30 - 2 + 5)/3]

⇒ (αβ + βγ + γα) = -27/3

⇒ (αβ + βγ + γα) = c/a

⇒ (αβγ) = 5(-2)(1/3)

⇒ (αβγ) = -10/3

⇒ (αβγ) = -d/a

Answer 2

Given :-

3x³ - 10x² - 27x + 10

To Find :-

Relation between its zeroes and coefdicients .​

Solution :-

First we will put the value of x in the equation and if the result come 0. Then, it will be a zero or more or less than 0 then, it is not a zero

For 5

p(x) = 3x³ - 10x² - 27x + 10

0 = 3(5)³ - 10(5)² - 27(5) + 10

0 = 3(125) - 10(25) - 135 + 10

0 = 375 - 250 - 135 + 10

0 = 385 - 385

0 = 0

For -2

p(x) = 3x³ - 10x² - 27x + 10

0 = 3(-2)³ - 10(-2)² - 27(-2) + 10

0 = 3(-8) - 10(4) - (-54) + 10

0 = -24 - 40 + 54 + 10

0 = -64 + 64

0 = 0

For 1/3

p(x) = 3x³ - 10x² - 27x + 10

0 = 3(1/3)³ - 10(1/3)² - 27(1/3) + 10

0 = 3 × 1/27 - 10 × 1/9 - 27 × 1/3 + 10

0 = 3/27 - 10/9 - 27/3 + 10

0 = 0

Now

αβγ = -(b)/a

5 + (-2) + 1/3 = -(-10)/3

5 - 2 + 1/3 = 10/3

15 - 6 + 1/3 = 10/3

10/3 = 10/3

Verified

αβ + βγ + γα = c/a

(5)(-2) + (-2)(1/3) + (1/3)(5) = -27/3

-10 + (-2/3) + (5/3) = -9

-10 - 2/3 + 5/3 = -9

-30 - 2 + 5/3 = -9

-32 + 5/3 = -9

-27/3 = -9

-9 = -9

Verified

αβγ = -(d)/a

(5)(-2)(1/3) = -(10)/3

-10 × 1/3 = -10/3

-10/3 = -10/3

Verified