Question

verify 1/2 , 1, -2 are zeros of 2x cube + x square -5x + 2 and check relationship between zeroes and coefficients​

Answer 1

EXPLANATION.

1/2,1,-2 are the zeroes of the quadratic equation,

Equation ⇒ 2x³ + x² - 5x + 2.

Let α = 1/2  and  β = 1  and  γ = -2.

Sum of zeroes of the cubic equation.

⇒ α + β + γ = -b/a.

⇒ 1/2 + 1 + (-2).

⇒ 1/2 + 1 - 2.

⇒ 1 + 2 - 4/2.

⇒ 3 - 4/2.

⇒ -1/2 = -b/a.

Products of zeroes of the cubic equation,

αβγ = -d/a.

⇒ (1/2)(1)(-2).

⇒ -1. = -d/a

Products of zeroes of cubic equation, two at a time.

αβ + βγ + γα = c/a.

⇒ (1/2)(1) + (1)(-2) + (-2)(1/2).

⇒ 1/2 - 2 - 1.

⇒ 1 - 4 - 2/2.

⇒ -5/2 = c/a.

Again, As we know that,

Put x = 1/2 in equation, we get.

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

⇒ p(1/2) = 1/4 + 1/4 - 5/2 + 2.

⇒ p(1/2) = 1 + 1 - 10 + 8/4.

⇒ p(1/2) = 10 - 10/4.

⇒ p(1/2) =0.

Put x= 1 in equation, we get.

⇒ p(1) = 2(1)³ + (1)² - 5(1) + 2.

⇒ p(1) = 2 + 1 - 5 + 2.

⇒ p(1) = 5 - 5.

⇒ p(1) = 0.

Put x = -2 in equation, we get.

⇒ p(-2) = 2(-2)³ + (-2)² - 5(-2) + 2.

⇒ p(-2) = -16 + 4 + 10 + 2.

⇒ p(-2) = 16 - 16.

⇒ p(-2) = 0.

HENCE VERIFIED.

Answer 2

$\sf{Answer}$

Step by step explanation:-

Given :-

Cubic equation :- 2x³+x²-5x+2

To find :-

Relationship between zeroes and coefficients

To verify :-

½ , 1 , -2 are zeroes of cubic equation

Solution :-

First we will verify zeroes of cubic equation

2x³ + x² -5x +2 =0

Put x = 1/2

2 × (1/2)³ + (1/2)² -5(1/2) + 2 = 0

2 × 1/8 + 1/4 - 5/2 +2 =0

1/4 + 1/4 -5/2 +2 =0

2/4 -5/2 + 2 =0

1/2 -5/2 + 2 =0

-4/2 + 2 =0

0/2 =0

0 = 0

So, p(1/2) =0

Hence p(1/2) is a factor

_______________

2x³ + x² -5x +2 =0

p(1) =

Put x = 1

2(1)³ +(1)² -5(1) +2 =0

2 + 1 -5 + 2 =0

5 -5 =0

0 =0

Hence p(1) is a factor

_________________________

2x³+x²-5x+2 = 0

p(-2)

Put x = -2

2(-2)³ + (-2)² -5(-2) +2 =0

-16 + 4 + 10 + 2 =0

16 -16 =0

0 = 0

Hence p(-2) is a factor

______________________________

From these we can say that

½ , 1 , -2 are zeroes of cubic equation

______________________________

Relation between zeroes & coefficients in cubic equation

We know Some relations:-

Sum of zeroes = α + ß + γ = -b/a

Sum of product of roots = α ß + ß γ + γα = c/a

Product of roots αßγ =-d/a

Now lets see !!

2x³+x²-5x+2 = 0

a = 2

b = 1

c = -5

d = 2

Sum of zeroes = -b/a = -1/2

Sum of product of roots = c/a = -5/2

product of roots = -d/a = -2/2 = -1

Hope u helpful

Thank u :)