Question

If x = 1 and x = −1 are the zeroes of the polynomial f(x) = 2x^3 − 5x^2 + ax + b,  find the value of 4ab.​

Answer 1

Answer:-

The Required Value Of 4ab is -40.

Explanation:-

Given Polynomial:-

$\\ \pink{ \leadsto \boxed{ \orange{ \bf p(x) = 2 {x}^{3} - 5 {x}^{2} + ax + b.}}}$

And two zeroes of the p(x) are 1 and -1.

To Find:-

• The value of 4ab.

Now,

It is given that 1 and -1 are zeroes of the polynomial so if we put 1 and -1 in place of x in the given p(x) then the result must be equal to 0.

Therefore,

Case I:-

▪︎ When x is 1.

$\\ \bf\mapsto p(1) = 0.$

$\\ \tt\mapsto 2(1) {}^{3} - 5(1) {}^{2} + a(1) + b = 0.$

$\\ \tt\mapsto2 - 5 + a + b =0.$

$\\ \tt\mapsto - 3 + a + b = 0.$

$\\ \tt\mapsto a + b = 0 + 3.$

$\\ \large\orange{\mapsto \boxed{ \red{ \bf a + b = 3.}}}...(1)$

Case II:-

▪︎ When x is -1.

$\\ \bf\mapsto p( - 1) = 0.$

$\\ \tt\mapsto 2( - 1) {}^{3} - 5( - 1) {}^{2} + a( - 1) + b = 0.$

$\\ \tt\mapsto - 2 + 5 - a + b = 0.$

$\\ \tt\mapsto2 ( - 1) - 5(1) + a( - 1) + b = 0.$

$\\ \tt\mapsto - 2 - 5 - a + b = 0.$

$\\ \tt\mapsto - 7 - a + b = 0.$

$\\ \tt\mapsto - a + b = 0 + 7.$

$\\ \large\orange{\mapsto \boxed{ \red{ \bf - a + b = 7.}}}...(2)$

By Adding eq(1) and eq(2) we get:-

$\\ \tt\mapsto(a + b) + ( - a + b) = 3 + 7.$

$\\ \tt\mapsto \cancel{a} + b \: \: \cancel{ - a} + b = 10.$

$\\ \tt\mapsto2b = 10.$

$\\ \tt\mapsto b = \cancel\dfrac{10}{2} .$

$\\ \large\red{\mapsto \boxed{ \blue{ \bf b = 5.}}}$

So the value of b is 5.

By putting the value of b in eq(1) we get:-

$\\ \tt\mapsto a + b = 3.$

$\\ \tt\mapsto a + 5 = 3.$

$\\ \tt\mapsto a = 3 - 5.$

$\\ \large\red{\mapsto \boxed{ \blue{ \bf a= - 2.}}}$

So The Value Of a is -2.

Therefore,

$\\ \pink{ \mapsto \boxed{ \red{ \bf4ab = 4( - 2)(5) = - 40.}}}$

Therefore The value of 4ab is -40.