Sure, let's solve and simplify the expression [tex]\( x^3 + x^5 - x^7 \)[/tex].
To simplify the given expression, we need to look for common factors in each term.
The expression is:
[tex]\[ x^3 + x^5 - x^7 \][/tex]
1. First, identify the greatest common factor (GCF) of all the terms:
- The GCF of [tex]\( x^3 \)[/tex], [tex]\( x^5 \)[/tex], and [tex]\( x^7 \)[/tex] is [tex]\( x^3 \)[/tex], as it is the highest power of [tex]\( x \)[/tex] that can be factored out from each term.
2. Factor [tex]\( x^3 \)[/tex] out from each term:
[tex]\[ x^3 (1) + x^3 (x^2) - x^3 (x^4) \][/tex]
3. Simplify to:
[tex]\[ x^3 \left( 1 + x^2 - x^4 \right) \][/tex]
After factoring, we get:
[tex]\[ x^3 \left( 1 + x^2 - x^4 \right) \][/tex]
However, if we reconsider the expanded form, we observe that the simplest expression without further reduction is:
[tex]\[ -x^7 + x^5 + x^3 \][/tex]
Thus, the simplified expression for [tex]\( x^3 + x^5 - x^7 \)[/tex] remains:
[tex]\[ -x^7 + x^5 + x^3 \][/tex]
So the simplified form of [tex]\( x^3 + x^5 - x^7 \)[/tex] is:
[tex]\[ -x^7 + x^5 + x^3 \][/tex]
