Question

How do you factor `x^5+5x^4+6x^3`?

Answer 1

`x^5+5x^4+6x^3 = x^3(x+2)(x+3)`

Explanation:

First note that all of the terms are divisible by `x^3`, so we can separate that out as a factor first:

`x^5+5x^4+6x^3 = x^3(x^2+5x+6)`

To factor the remaining quadratic, we want to find two numbers whose sum is `5` and whose product is `6`, since in general we have:

`(x+p)(x+q) = x^2+(p+q)x+pqx`

The numbers `2, 3` work, so we have:

`x^2+5x+6 = (x+2)(x+3)`

Putting it all together:

`x^5+5x^4+6x^3 = x^3(x+2)(x+3)`

Answer 2

`x^3(x+3)(x+2)`

Explanation:

Factorization is accomplished by taking a common factor ,completing the square, applying the polynomial identities ,or using the quadratic formula .

`x^5+5x^4+6x^3`
`=color(blue)(x^3)color(brown)((x^2+5x+6)`

The expression `x^2+5x+6` can be factorized using the trial and error method, in other words, the idea is to find two integers whose sum is `color(red) 5` and their product is `color(violet) 6`.

`X^2+color(red)SX+color(violet)P`

For the expression `x^2+5x+6` , the two integers satisfying the `color(red)S and color(violet)P` requirements are `3 and 2`.

`color(brown)(x^2+5x+6=(x+3)(x+2))`

For the whole expression`\ x^5+5x^4+6x^3\`, the factorized form is:

`color(blue)(x^3)color(brown)((x+3)(x+2)`

Therefore,

`x^5+5x^4+6x^3=x^3(x+3)(x+2)`