To find the greatest common factor (GCF) of the expression [tex]\( 7x^3a + 7x^2a^2 \)[/tex], we will follow these steps:
1. Identify the common factors in each term:
- The first term is [tex]\( 7x^3a \)[/tex].
- The numerical coefficient is [tex]\( 7 \)[/tex].
- The variable factors are [tex]\( x^3 \)[/tex] and [tex]\( a \)[/tex].
- The second term is [tex]\( 7x^2a^2 \)[/tex].
- The numerical coefficient is [tex]\( 7 \)[/tex].
- The variable factors are [tex]\( x^2 \)[/tex] and [tex]\( a^2 \)[/tex].
2. Find the greatest common factor of the coefficients:
- Both terms have the numerical coefficient [tex]\( 7 \)[/tex].
3. Identify the lowest power of each variable present in all terms:
- Both terms include the variable [tex]\( x \)[/tex].
- The first term has [tex]\( x^3 \)[/tex].
- The second term has [tex]\( x^2 \)[/tex].
- The lowest power of [tex]\( x \)[/tex] common to both terms is [tex]\( x^2 \)[/tex].
- Both terms include the variable [tex]\( a \)[/tex].
- The first term has [tex]\( a \)[/tex] (which can be written as [tex]\( a^1 \)[/tex]).
- The second term has [tex]\( a^2 \)[/tex].
- The lowest power of [tex]\( a \)[/tex] common to both terms is [tex]\( a \)[/tex] or [tex]\( a^1 \)[/tex].
4. Combine these common factors:
- The numerical common factor is [tex]\( 7 \)[/tex].
- The greatest common factor of [tex]\( x \)[/tex] is [tex]\( x^2 \)[/tex].
- The greatest common factor of [tex]\( a \)[/tex] is [tex]\( a \)[/tex].
Putting it all together, the greatest common factor of the expression [tex]\( 7x^3a + 7x^2a^2 \)[/tex] is:
[tex]$
7a x^2
$[/tex]
