Question

Find the zeroes of the polynomial 4x^2+5√2x-3 and verify the relationship between the zeroes and the coefficients of the polynomial.

Answer 1

Finding zeores :-

p(x) = 4x^2 + 5✓2x - 3

( splitting the middle term )

= 4x^2 + 6✓2x - ✓2x - 3

= 2✓2x (✓2x + 3) - 1(✓2x + 3)

= (✓2x + 3) (2✓2x - 1) = 0

= (✓2x + 3) = 0 and (2✓2x - 1) = 0

x = -3/✓2 OR x = 1/2✓2

Therefore the zeroes are -3/✓2 and 1/2✓2

Verifying Relationship between the zeroes and Coefficients :-

α = -3/✓2

β = 1/2✓2

Sum of Zeros= (α + β) = -3/✓2 + 1/2✓2 = -3×2✓2 + ✓2 = -6✓2+✓2/4 = -5✓2/4

Product of zeros = (αβ) -3/✓2 × 1/2✓2 = -3/4

Answer 2

Answer:

Step-by-step explanation:

Solution :-

We have polynomial is P(X) = 4X²+5✓2X-3

Solving all the values

⇒ 4X²+6✓2X-✓2X-3

⇒ 2✓2X(✓2X+3) -1(✓2X+3)

⇒ (✓2X+3) (2✓2X-1) = 0

⇒ (✓2X+3) = 0 OR (2✓2X-1) = 0

⇒ X = -3/✓2 OR X = 1/2✓2

-3/✓2 and 1/2✓2 are the two zeros of the given polynomial.

Checking relationship between the zeroes and Coefficients

Let Alpha = -3/✓2 and beta = 1/2✓2

Sum of Zeros = (Alpha + Beta) = -3/✓2 + 1/2✓2 = -3×2✓2 + ✓2 = -6✓2+✓2/4 = -5✓2/4 = -( Coefficient of X/Coefficient of X².

Product of zeros = (-3/✓2 × 1/2✓2) = -3/4 = Constant term/Coefficient of X².