Answer :
To find the greatest common factor (GCF) of [tex]\( 14x^2 \)[/tex] and [tex]\( 9x^4 \)[/tex], we will follow these steps:
1. Factorize each term:
- [tex]\( 14x^2 \)[/tex]:
- [tex]\( 14 \)[/tex] can be factorized into prime factors: [tex]\( 14 = 2 \times 7 \)[/tex].
- [tex]\( x^2 \)[/tex] is already factorized as [tex]\( x \times x \)[/tex].
So, [tex]\( 14x^2 = 2 \times 7 \times x^2 \)[/tex].
- [tex]\( 9x^4 \)[/tex]:
- [tex]\( 9 \)[/tex] can be factorized into prime factors: [tex]\( 9 = 3 \times 3 = 3^2 \)[/tex].
- [tex]\( x^4 \)[/tex] is already factorized as [tex]\( x \times x \times x \times x = x^4 \)[/tex].
So, [tex]\( 9x^4 = 3^2 \times x^4 \)[/tex].
2. Identify the common factors:
- For the numerical coefficients:
- The factors of 14 (which are 2 and 7) do not have any common factors with the factors of 9 (which are [tex]\( 3^2 \)[/tex]). Therefore, there is no common numerical factor other than 1.
- For the variable part:
- Look at the powers of [tex]\( x \)[/tex]. The smallest power of [tex]\( x \)[/tex] that appears in both expressions is [tex]\( x^2 \)[/tex].
3. Combine the common factors:
Since the numerical part has no common factors, we focus on the variable part.
The greatest common factor of [tex]\( 14x^2 \)[/tex] and [tex]\( 9x^4 \)[/tex] is [tex]\( x^2 \)[/tex].
So, the greatest common factor is [tex]\( \boxed{x^2} \)[/tex].
1. Factorize each term:
- [tex]\( 14x^2 \)[/tex]:
- [tex]\( 14 \)[/tex] can be factorized into prime factors: [tex]\( 14 = 2 \times 7 \)[/tex].
- [tex]\( x^2 \)[/tex] is already factorized as [tex]\( x \times x \)[/tex].
So, [tex]\( 14x^2 = 2 \times 7 \times x^2 \)[/tex].
- [tex]\( 9x^4 \)[/tex]:
- [tex]\( 9 \)[/tex] can be factorized into prime factors: [tex]\( 9 = 3 \times 3 = 3^2 \)[/tex].
- [tex]\( x^4 \)[/tex] is already factorized as [tex]\( x \times x \times x \times x = x^4 \)[/tex].
So, [tex]\( 9x^4 = 3^2 \times x^4 \)[/tex].
2. Identify the common factors:
- For the numerical coefficients:
- The factors of 14 (which are 2 and 7) do not have any common factors with the factors of 9 (which are [tex]\( 3^2 \)[/tex]). Therefore, there is no common numerical factor other than 1.
- For the variable part:
- Look at the powers of [tex]\( x \)[/tex]. The smallest power of [tex]\( x \)[/tex] that appears in both expressions is [tex]\( x^2 \)[/tex].
3. Combine the common factors:
Since the numerical part has no common factors, we focus on the variable part.
The greatest common factor of [tex]\( 14x^2 \)[/tex] and [tex]\( 9x^4 \)[/tex] is [tex]\( x^2 \)[/tex].
So, the greatest common factor is [tex]\( \boxed{x^2} \)[/tex].
