Question

x cube + 2 x square - 5 x minus 6

Answer 1

x^3 +2x^2 -5x -6
put x =-1
the eq will be zero it's mean x+1 is it's factor
(x+1)(x^2 +x-6)
(x+1)(x^2+3x-2x-6)
(x+1)(x+3)(x-2) Ans

Answer 2

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[Factorize x cube + 2 x square - 5 x minus 6]

Concept

Euclidean division lemma (the lemma is similar to a theorem) states that given two positive integers "a" and "b", there exist unique integers "q" and "r" such that is: a = bq+ r, where 0 ≤ r < b.

where "a" is dividend , "b" is divisor , "q" is quotient and" r" is remainder

Given

We have been given a cubic equation $x^3+2x^2-5x-6$

Find

We are asked to determine the factors of the given equation .

Solution

It is given that  $x^3+2x^2-5x-6=0$ .    ...(1)

Putting x = -1 in equation, we get

$(-1)^3+2(-1)^2-5(-1)-6=-1+2+5-6=0$

Hence, on putting x = -1 the equation gives zero which means that x = -1 is the solution of the given equation which can be written as

$x=-1\\x+1=0$    ...(2)

On dividing equation (1) by equation (2), we get quotient

$\frac{x^3+2x^2-5x-6}{x+1} =x^2+x-6$

By using Division lemma which is given by

$Dividend =Divisor \times Quotient+Remainder$ ...(3)

Putting Dividend $=x^3+2x^2-5x-6$ , Divisor $=x+1$, Quotient $x^2+x-6$ , Remainder = 0 in equation (3) , we get

$x^3+2x^2-5x-6=(x+1)(x^2+x-6)+0$ ...(4)

We will factorize the quotient

$x^2+x-6=0\\x^2+(3-2)x-6=0\\x^2+3x-2x-6=0\\x(x+3)-2(x+3)=0\\(x-2)(x+3)=0$

Putting this in equation (4) , we get

$x^3+2x^2-5x-6=(x+1)(x-2)(x+3)$

Therefore , the factore of given equation is (x+1)(x-2)(x+3).

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