To factorize the expression [tex]\( x^2 - 22x + 121 \)[/tex] in the form [tex]\( (x - A)^B \)[/tex], let's follow a series of logical steps:
1. Identify the quadratic expression: We start with the given quadratic expression:
[tex]\[
x^2 - 22x + 121.
\][/tex]
2. Recognize the form of a perfect square: A perfect square trinomial can be written in the form [tex]\( (x - p)^2 \)[/tex] or [tex]\( (x + p)^2 \)[/tex]. In our given expression, we note that [tex]\( 121 \)[/tex] is a perfect square of [tex]\( 11 \)[/tex], i.e., [tex]\( 11^2 = 121 \)[/tex].
3. Verify the middle term: To determine if the expression is a perfect square trinomial, we check if the middle term (in this case, [tex]\(-22x\)[/tex]) can be written as [tex]\( -2 \cdot 11 \cdot x \)[/tex]. We see that:
[tex]\[
-2 \cdot 11 \cdot x = -22x.
\][/tex]
4. Write the expression as a square of a binomial: Since both conditions fit (i.e., the last term is a perfect square and the middle term is twice the product of the square root of the last term and [tex]\( x \)[/tex]), we can write the expression as:
[tex]\[
x^2 - 22x + 121 = (x - 11)^2.
\][/tex]
Now, having [tex]\( (x - A)^B \)[/tex] from the problem statement, we can identify the constants [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- The expression inside the parentheses is [tex]\( (x - 11) \)[/tex], which indicates [tex]\( A = 11 \)[/tex].
- The exponent outside is [tex]\( 2 \)[/tex], which means [tex]\( B = 2 \)[/tex].
Thus:
[tex]\[
x^2 - 22x + 121 = (x - 11)^2,
\][/tex]
with:
[tex]\[
A = 11,
\][/tex]
[tex]\[
B = 2.
\][/tex]
