Question

How do i solve this question, please help and pls explain
a³x - x⁴ + a²x² - ax³

Answer 1

`\huge\boxed{x(a-x)(a+x)^2}`

Explanation:

In order to factor this expression, our goal is to write the expression in a way that we can factor out a term.

With the expression
`a^3 x - x^4 + a^2 x^2 - ax^3`, we need to note an exponent rule.

`a^(b+c) = a^b a^c`

Step 1:

We can use this to get each term of this expression to have a term of
`x` so we can factor it out.

Let's look at each term and get it so that we can factor out an x term.

• 
  `a^3 x = x a^3` (Commutative Property)
• 
  `x^4 = xx^3` (Since
  `x^3 \cdot x^1 = x^4`)
• 
  `a^2 x^2 = a^2 \cdot xx` (Since
  `x^2 = x \cdot x`)
• 
  `ax^3 = a \cdot xx^2` (Since
  `x^3 = x^2 \cdot x^1`

With this, our equation becomes
`(xa^3) - (xx^3) + (xxa^2) - (xx^2a)`.

We now can factor out the common term x.

`x(a^3 - x^3 + xa^2 - x^2a)`

Step 2:

From here, we can now factor
`a^3 - x^3 + xa^2 - x^2 a`

• Rearrange the equation:
  `a^3 +xa^2 - x^3 - x^2a`
• Factor out
  `a^2` from
  `a^3 + xa^2` which comes out to be
  `a^2(a+x)`
• Factor out
  `-x^2` from
  `-x^3 - x^2a` which comes out to be
  `-x^2(x+a)`
• We now have
  `a^2(a+x) - x^2(x+a)`
• Factor out the common term,
  `(a+x)`, which comes out to be
  `(a+x)(a^2 - x^2)`
• Factor
  `-x^2+a^2` into
  `(a+x)(a-x)`

• We now have
  `(a+x) (a+x) (a-x)`, which is simplified to
  `(a-x)(a+x)^2`

Finalizing:

Since we have just factored
`a^3 - x^3 + xa^2 - x^2 a` and factored x out of
`a^3 x - x^4 + a^2 x^2 - ax^3` in the first couple of steps, we need to have it as a factorization of x.

`x(a-x)(a+x)^2`

Hope this helped!