Question

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

Answer 1

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.

$a^{b+c} = a^b a^c$

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!