Sure! Let's factor the expression step-by-step.
Given the expression:
[tex]\[ x(a^2 - b^2) + a(b^2 - x^2) \][/tex]
First, let's recall some useful algebraic identities and distributive properties. One key identity is the difference of squares:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Now, we need to rewrite and factor the given expression:
[tex]\[ x(a^2 - b^2) + a(b^2 - x^2) \][/tex]
1. Step 1: Apply the difference of squares
Begin by expanding each term using the difference of squares where possible:
[tex]\[ x(a^2 - b^2) = x(a - b)(a + b) \][/tex]
[tex]\[ a(b^2 - x^2) = a(b - x)(b + x) \][/tex]
Notice that we could write the second term as [tex]\( a(- (x^2 - b^2)) = -a(x^2 - b^2) = -a(x - b)(x + b) \)[/tex].
2. Step 2: Substitute these results back into the original expression
This gives us:
[tex]\[
x(a - b)(a + b) + a(- (x - b)(x + b))
\][/tex]
This can be rewritten as:
[tex]\[
x(a - b)(a + b) - a(x - b)(x + b)
\][/tex]
3. Step 3: Notice the common factor
Observe that [tex]\( (a - b) \)[/tex] and [tex]\( (x + b) \)[/tex] appear in both terms, so we can factor these terms out:
[tex]\[
(a - b) \left[x(a + b) - a(x + b)\right]
\][/tex]
4. Step 4: Simplify inside the brackets
Simplify the expression inside the brackets:
[tex]\[
x(a + b) - a(x + b)
\][/tex]
Distribute [tex]\( x \)[/tex] and [tex]\( -a \)[/tex]:
[tex]\[
xa + xb - ax - ab = xa - ax + xb - ab
\][/tex]
Since terms [tex]\( xa \)[/tex] and [tex]\( -ax \)[/tex] cancel each other out, we're left with:
[tex]\[
xb - ab = (x - a)b
\][/tex]
So, combining these simplifications, we have:
[tex]\[
(a - x)(a x + b^2)
\][/tex]
Therefore, the final factored form of the given expression is:
[tex]\[
-\left(-a + x\right)\left(a x + b^2\right)
\][/tex]
And that completes the factorization!
