Question

for the polynomial 2x^3 - 5x^2 -14x +8, find the sum of the products of the zeroes taken two at a time a. 7 b. 8 c. -7 d. -8 solve

Answer 1

Solution :-

First we have to factorize the given cubic expression 2x³ - 5x² - 14x + 8..

By Hit & Trial , when we put x = (-2) ,

→ p(x) = 2x³ - 5x² - 14x + 8

→ p(-2) = 2(-2)³ - 5(-2)² - 14(-2) + 8

→ p(-2) = 2*(-8) -5*4 +28 + 8

→ p(-2) = (-16) - (20) + 36

→ p(-2) = (-36) + 36

→ p(-2) = 0

So, we can conclude that, (x + 2) is a factor of given Polynomial .

Now,

→ 2x³ - 5x² - 14x + 8

→ 2x³ + 4x² - 9x² - 18x + 4x + 8

→ 2x²(x + 2) - 9x(x + 2) + 4(x + 2)

→ (x + 2)(2x² - 9x + 4)

→ (x + 2)(2x² - 8x - x + 4)

→ (x + 2)[2x(x - 4) - 1(x - 4)]

→ (x + 2)(x - 4)(2x - 1)

Putting All Equals to Zero Now,

→ x = (-2) , 4 and (1/2) = Zeros of given Cubic Polynomial.

Therefore,

→ Sum of the Products of the zeroes taken two at a time =

→ (-2)*4 + 4*(1/2) + (1/2)*(-2)

→ (-8) + 2 + (-1)

→ (-9) + 2

→ (-7) (Option C) (Ans.)

Answer 2

Given:

• A cubic polynomial is given to us.
• The polynomial is 2x³-5x²-14x+8.

To Find:

• The sum of the products of the zeroes taken two at a time .

Answer:

Of a cubic polynomial in standard form ax³+bx²+cx+d , the sum of the products of the zeroes taken two at a time is given by

$\large{\underline{\boxed{\red{\bf{\leadsto \alpha\beta+\beta\gamma+\gamma\alpha = \dfrac{c}{a}}}}}}$

• Where alpha , beta and gamma are the zeroes of the Cubic polynomial.

Now the given polynomial ,

• c = (-14)
• a = 2 .

Hence on putting respective values,

= (-14)/2

= (-7)

Hence the required answer is -7.