To simplify the given expression [tex]\(\frac{a^4}{b^4} - \frac{23 a^2}{b^2} + 45\)[/tex], let's consider it in terms of [tex]\(u\)[/tex], where [tex]\(u = \frac{a^2}{b^2}\)[/tex]. This substitution helps in simplifying the polynomial expression.
First, rewrite the expression using [tex]\(u\)[/tex]:
[tex]\[ \frac{a^4}{b^4} - \frac{23 a^2}{b^2} + 45 \][/tex]
Since [tex]\(\frac{a^4}{b^4} = \left(\frac{a^2}{b^2}\right)^2\)[/tex], and [tex]\(\frac{a^2}{b^2} = u\)[/tex], we can rewrite the expression as:
[tex]\[ u^2 - 23u + 45 \][/tex]
So, the given expression simplifies to a quadratic polynomial in terms of [tex]\(u\)[/tex]:
[tex]\[ u^2 - 23u + 45 \][/tex]
Now, let's try to factorize this quadratic expression, if possible. To factorize [tex]\(u^2 - 23u + 45\)[/tex], we need to find two numbers that multiply to 45 and add up to -23.
After some thought, we find that the numbers are -3 and -15, since:
[tex]\[ (-3) \cdot (-15) = 45 \][/tex]
[tex]\[ (-3) + (-15) = -18 \][/tex]
We see our middle coefficient is -23, not -18, meaning our initial attempt to factorize has a mistake due to typo or intention of specifically picking a quadratic that can't neatly split into integers, reconsidering back the expression then:
So the simplified form remains [tex]\( u^2 - 23u + 45 \)[/tex]
Substituting back [tex]\(u\)[/tex]:
[tex]\[ \left(\frac{a^2}{b^2}\right)^2 - 23\left(\frac{a^2}{b^2}\right) + 45 \][/tex]
Therefore, the fully simplified and correct form of the original expression is:
[tex]\[ \frac{a^4}{b^4} - \frac{23 a^2}{b^2} + 45 \][/tex]
This is the simplified result of the given mathematical expression.
