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TEA [102]
3 years ago
9

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

Mathematics
1 answer:
taurus [48]3 years ago
7 0

Answer:

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

Step-by-step 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

<h2><u><em>Step 1:</em></u></h2>

<u><em></em></u>

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 <em>(Commutative Property)</em>
  • x^4 = xx^3 <em>(Since </em>x^3 \cdot x^1 = x^4)
  • a^2 x^2 = a^2 \cdot xx <em>(Since </em>x^2 = x \cdot x)
  • ax^3 = a \cdot xx^2 <em>(Since </em>x^3 = x^2 \cdot x^1<em />

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)

<h2><u><em>Step 2:</em></u></h2>

<u><em></em></u>

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

<h2><u><em>Finalizing:</em></u></h2>

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!

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