Question

verify that 3,-1,- 1/3 are the zeros of the
polynomial 3x³-5x²-11x-3 and check
the relationship between the zeros of and
Cofficints!​

Answer 1

EXPLANATION.

Verify that 3,-1,-1/3 are the zeroes of the cubic polynomial 3x³ - 5x² - 11x - 3.

Check the relationship between the zeroes and coefficients.

Put the value of x = 3 in equation,

⇒ p(x) = 3x³ - 5x² - 11x - 3.

⇒ p(3) = 3(3)³ - 5(3)² - 11(3) - 3.

⇒ p(3) = 81 - 45 - 33 - 3.

⇒ p(3) = 81 - 81.

⇒ p(3) = 0.

Put the value of x = -1 in equation,

⇒ p(-1) = 3(-1)³ - 5(-1)² - 11(-1) - 3.

⇒ p(-1) = -3 - 5 + 11 - 3.

⇒ p(-1) = 11 - 11.

⇒ p(-1) = 0.

Put the value of x = -1/3 in equation,

⇒ p(-1/3) = 3(-1/3)³ - 5(-1/3)² - 11(-1/3) - 3.

⇒ p(-1/3) = -1/9 - 5/9 + 11/3 - 3.

⇒ p(-1/3) = -1 - 5 + 33 - 27/9.

⇒ p(-1/3) = 33 - 33/9.

⇒ p(-1/3) = 0.

Cubic Polynomial = x³ - (α + β + γ)x² + (αβ + βγ + γα)x - αβγ.

Sum of zeroes of cubic polynomial,

⇒ α + β + γ = -b/a.

⇒ α + β + γ = -(-5)/3 = 5/3.

Product of zeroes of cubic polynomial,

⇒ αβγ = -d/a.

⇒ αβγ = -(-3)/3 = 1.

Product of zeroes of cubic polynomial two at a time,

⇒ αβ + βγ + γα = c/a.

⇒ αβ + βγ + γα = -11/3.

Put it on a equation we get,

⇒ x³ - (5/3)x² + (-11/3)x - 1 = 0.

⇒ 3x³ - 5x² - 11x - 3/3 = 0.

⇒ 3x³ - 5x² - 11x - 3 = 0.

HENCE PROVED.

Answer 2

$\Huge \fbox \red{an} \fbox \green{s} \fbox \pink{we} \fbox \purple{r}$

Let given cubic polynomial be

p(x) = 3x³ - 5x² - 11x - 3

i ) If x = 3 , then

p(3) = 3(3)³ - 5(3)² - 11(3) - 3

= 81 - 45 - 33 - 3

= 81 - 81

= 0

ii ) If x = -1 , then

p(-1) = 3(-1)³ - 5(-1)² - 11(-1) - 3

= -3 - 5 + 11 - 3

= -11 + 11

= 0

iii ) If x = -1/3 , then

p(-1/3) = 3(-1/3)³ - 5(-1/3)² - 11(-1/3) - 3

= -3/27 - 5/9 + 11/3 - 3

= -1/9 - 5/9 + 11/3 - 3

= ( -1 - 5 + 33 - 27 )/9

= ( -33 + 33 )/9

= 0

Therefore ,

p(3) = p(-1) = p(-1/3) = 0 .

So, 3 , -1 , -1/3 are zeroes of given

cubic polynomial 3x³ - 5x² - 11x - 3 .

Compare the coefficients of given

cubic polynomial 3x³ - 5x² - 11x - 3

with ax³ + bx² + cx + d , we get

a = 3 , b = -5 , c = -11 , d = -3

Let the zeroes of the given cubic

polynomial p(x) are p =3 , q=-1, r=-1/3

Now ,

p+q+r = 3 - 1 - 1/3

= 2 - 1/3

= ( 6 - 1 )/3

= 5/3

= -b/a

pq+qr+rp

= 3(-1)+(-1)(-1/3)+(-1/3)(3)

= -3 + 1/3 - 1

= -4 + 1/3

= ( -12 + 1 )/3

= -11/3

= c/a

pqr = 3 × ( -1 ) × ( -1/3 )

= 1

=- d/a

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$\Huge \boxed{ \colorbox{lime}{hope \: this \: helps}}$