Question

Factories x cube minus 6 x + 3x + 10

Answer 1

Answer:

(x + 1)(x - 5)(x - 2)

Note:

★ The possible values of the variable for which the polynomial becomes zero are called its zeros .

★ A cubic polynomial can have atmost two roots .

★ If the polynomial becomes zero at x = a ( ie ; if p(a) = 0 ) , then x = a is a zero of the polynomial p(x) and hence (x - a) is a factor of the polynomial p(x) .

Solution:

Here ,

We need to factorize the given cubic polynomial is ; x³ - 6x² + 3x + 10 .

Thus,

Here we go ↓

By HIT AND TRIAL method we get that ,

x = -1 is a zero of the given cubic polynomial , ie ; the polynomial becomes zero at x = -1 .

Thus,

If x = -1 , then

x + 1 = 0

Hence ,

(x + 1) is a factor of the given cubic polynomial .

Now,

Let's divide the given cubic polynomial by (x + 1) .

x + 1 ) x³ - 6x² + 3x + 10 ( x² - 7x + 10

x³ + x²

– –

– 7x² + 3x

– 7x² – 7x

+ +

10x + 10

10x + 10

– –

0 0

Here ,

Dividend = x³ - 6x² + 3x + 10

Divisor = x + 1

Quotient = x² - 7x + 10

Remainder = 0

Also ,

We know that ;

Dividend = Divisor×Quotient + Remainder

Thus ,

=> x³ - 6x² + 3x + 10 = (x +1)(x² - 7x +10) + 0

=> x³ - 6x² + 3x + 10 = (x +1)(x² - 7x + 10)

=> x³ - 6x² + 3x + 10 = (x +1)(x²- 5x -2x + 10)

=> x³ - 6x² + 3x + 10 = (x +1)[x(x-5) - 2(x+5)]

=> x³ - 6x² + 3x + 10 = (x +1)(x - 5)(x - 2)

Hence ,

Required answer is ;

(x + 1)(x - 5)(x - 2)