Question

if a and b are the zeroes of the polynomial 2y²+7y+5 write tha value of a+b+2b​

Answer 1

Correct Question

If a and b are the zeros of the polynomial 2y² + 7y +5,find the value of a + b + ab

SOLUTION

Let the given polynomial be p(y) = 2y² + 7y + 5

Given that a and b are the zeros of the above polynomial

To finD

Value of a + b + ab

Now,

Splitting the middle term,

$\sf \: p(y) = 2 {y}^{2} + 7y + 5 \\ \\ \longrightarrow \: \sf \: p(y) = 2y {}^{2} + 2y + 5y + 5 \\ \\ \longrightarrow \: \sf \: p(y) = 2y(y + 1) + 5(y + 1) \\ \\ \longrightarrow \: \boxed { \boxed{\sf \: p(y) = (2y + 5)(y + 1)}}$

From Factor Theorem,

$\sf \: p(y) = 0 \\ \\ \leadsto \: \sf \: (y +1 )(2y + 5) = 0 \\ \\ \leadsto \: \sf \: \: y = - 1 \: or \: - \dfrac{5}{2}$

Now,

$\sf \: a + b = - \dfrac{7}{2}$

$\sf \: ab = \dfrac{5}{2}$

Thus,

$\sf \: (a + b) + ab \\ \\ \longmapsto \: \sf \: - \dfrac{7}{2} + \dfrac{5}{2} \\ \\ \longmapsto \: \sf \: - \dfrac{ 2}{2} \sim \: - 1$

The value of (a + b) + ab is - 1

ALITER

We could have found the sum and product of zeros from the following relations and derive the required result

• Sum of Zeros = - y coefficient/y² coefficient

• Product of Zeros = constant term/y² coefficient

Thus,

a + b = - 7/2

ab = 5/2

Now,

a + b + ab

» (-7/2) + 5/2

» - 1

Answer 2

Correct question:

If 'a' and 'b' are the zeros of the polynomial 2y² + 7y + 5, write the value of a + b + ab​.

Answer:

Let's suppose ax² + bx + c were a quadratic polynomial.

$\sf The\ sum\ of\ its\ zeros\ would\ be\ given\ by\ \dfrac{-b}{a}.\\\\\\The\ product\ of\ its\ zeros\ would\ be\ given\ by\ \dfrac{c}{a}.$

We've been given that 'a' and 'b' are the zeros.

This means that:

$\sf a\ +\ b\ =\ \dfrac{-b}{a}\\\\\\a\ \times\ b\ =\ \dfrac{c}{a}$

From the equation,

• a = 2
• b = 7
• c = 5

Using these values to solve the sum and product,

$\sf a\ +\ b\ =\ \dfrac{-7}{2}\\\\\\a\ \times\ b\ =\ \dfrac{5}{2}$

Therefore, a + b + ab ⇒

$\sf \dfrac{-7}{2}\ +\ \dfrac{5}{a}\ =\ -1$

a + b + ab = -1