To factor out the greatest common factor from the expression [tex]\(6x^4 - 18x^3 + 30x^2\)[/tex], we can follow these steps:
1. Identify the coefficients and common factors of each term:
- The terms in the expression are [tex]\(6x^4\)[/tex], [tex]\(-18x^3\)[/tex], and [tex]\(30x^2\)[/tex].
- The coefficients are 6, -18, and 30.
- The greatest common factor (GCF) of the coefficients 6, 18, and 30 is 6.
2. Identify the common factor in the variables:
- For the variables [tex]\(x^4\)[/tex], [tex]\(x^3\)[/tex], and [tex]\(x^2\)[/tex], the smallest power of [tex]\(x\)[/tex] is [tex]\(x^2\)[/tex]. This is because [tex]\(x^2\)[/tex] is present in all three terms and is the lowest exponent among them.
3. Combine the GCF of the coefficients and the variables:
- The GCF of the entire expression is [tex]\(6x^2\)[/tex].
4. Factor out [tex]\(6x^2\)[/tex] from each term in the expression:
- For [tex]\(6x^4\)[/tex]: [tex]\[
6x^4 = 6x^2 \cdot x^2
\][/tex]
- For [tex]\(-18x^3\)[/tex]: [tex]\[
-18x^3 = 6x^2 \cdot (-3x)
\][/tex]
- For [tex]\(30x^2\)[/tex]: [tex]\[
30x^2 = 6x^2 \cdot 5
\][/tex]
5. Write the factored form:
- When you factor out [tex]\(6x^2\)[/tex], the expression can be rewritten as:
[tex]\[
6x^4 - 18x^3 + 30x^2 = 6x^2 (x^2 - 3x + 5)
\][/tex]
Therefore, the expression [tex]\(6x^4 - 18x^3 + 30x^2\)[/tex] factored by its greatest common factor is:
[tex]\[
\boxed{6x^2 (x^2 - 3x + 5)}
\][/tex]
