To factor the expression [tex]\( 25u^3 x^8 - 15u x^3 y^9 \)[/tex], we should follow these steps:
1. Identify the Greatest Common Factor (GCF):
- For the coefficients 25 and 15, the GCF is 5.
- For the variable [tex]\( u \)[/tex], the lowest power between [tex]\( u^3 \)[/tex] and [tex]\( u \)[/tex] is [tex]\( u \)[/tex]. Hence, [tex]\( u \)[/tex] is the GCF for [tex]\( u \)[/tex].
- For the variable [tex]\( x \)[/tex], the lowest power between [tex]\( x^8 \)[/tex] and [tex]\( x^3 \)[/tex] is [tex]\( x^3 \)[/tex]. Hence, [tex]\( x^3 \)[/tex] is the GCF for [tex]\( x \)[/tex].
- The variable [tex]\( y \)[/tex] only appears in the second term, so it does not contribute to the GCF.
Combining these, the overall GCF of the expression is [tex]\( 5u x^3 \)[/tex].
2. Factor out the GCF:
- Divide each term in the original expression by the GCF [tex]\( 5u x^3 \)[/tex].
[tex]\[
\frac{25u^3 x^8}{5u x^3} = 5u^2 x^5
\][/tex]
[tex]\[
\frac{15u x^3 y^9}{5u x^3} = 3 y^9
\][/tex]
After factoring out [tex]\( 5u x^3 \)[/tex], the expression inside the parentheses becomes [tex]\( 5u^2 x^5 - 3 y^9 \)[/tex].
3. Write the factored form:
- Putting it all together, the factored expression is:
[tex]\[
5u x^3 (5u^2 x^5 - 3 y^9)
\][/tex]
Thus, the factored form of the expression [tex]\( 25u^3 x^8 - 15u x^3 y^9 \)[/tex] is:
[tex]\[
5u x^3 (5u^2 x^5 - 3 y^9)
\][/tex]
