Of course, let's go through the step-by-step simplification of the given mathematical expression [tex]\((a+b)(b-a)\)[/tex].
### Step 1: Recognize the expression
We start with the given expression:
[tex]\[
(a + b)(b - a)
\][/tex]
### Step 2: Apply the distributive property
The distributive property states that [tex]\( (x + y)(z + w) = xz + xw + yz + yw \)[/tex]. Applying this to our expression, we get:
[tex]\[
(a + b)(b - a) = a \cdot b + a \cdot (-a) + b \cdot b + b \cdot (-a)
\][/tex]
### Step 3: Perform individual multiplications
Now, let's multiply each term:
[tex]\[
a \cdot b = ab
\][/tex]
[tex]\[
a \cdot (-a) = -a^2
\][/tex]
[tex]\[
b \cdot b = b^2
\][/tex]
[tex]\[
b \cdot (-a) = -ab
\][/tex]
### Step 4: Combine the products
Now, let's combine all the products:
[tex]\[
ab - a^2 + b^2 - ab
\][/tex]
### Step 5: Simplify the expression
Notice that we have [tex]\(ab\)[/tex] and [tex]\(-ab\)[/tex] in the expression. They cancel each other out:
[tex]\[
(ab - ab) - a^2 + b^2 = 0 - a^2 + b^2
\][/tex]
### Step 6: Write the final simplified expression
After canceling out [tex]\(ab - ab\)[/tex], we are left with:
[tex]\[
-a^2 + b^2
\][/tex]
### Final Result
Thus, the expression [tex]\((a+b)(b-a)\)[/tex] simplifies to:
[tex]\[
-a^2 + b^2
\][/tex]
