Answer :
To factor out the greatest common factor (GCF) in the expression [tex]7x^4 - 14x^3 + 21x^2[/tex], we first need to identify the largest factor that is common to each term in the expression.
Step 1: Identify the coefficients and variables in each term.
- The terms are [tex]7x^4[/tex], [tex]-14x^3[/tex], and [tex]21x^2[/tex].
- The coefficients are 7, -14, and 21.
Step 2: Find the GCF of the coefficients:
- Break down each coefficient into its prime factors:
- 7 is already a prime number.
- 14 breaks down to [tex]2 \times 7[/tex].
- 21 breaks down to [tex]3 \times 7[/tex].
- The common factor among these numbers is 7.
Step 3: Identify the variable factor common to each term:
- The smallest power of [tex]x[/tex] that appears in all terms is [tex]x^2[/tex].
Step 4: Combine these to find the GCF of the entire expression:
- The GCF of the expression is [tex]7x^2[/tex].
Step 5: Factor the GCF out from the expression:
- Divide each term by the GCF [tex]7x^2[/tex]:
- [tex]\frac{7x^4}{7x^2} = x^2[/tex]
- [tex]\frac{-14x^3}{7x^2} = -2x[/tex]
- [tex]\frac{21x^2}{7x^2} = 3[/tex]
- Write the factored expression:
[tex]7x^2(x^2 - 2x + 3)[/tex]
Therefore, the expression [tex]7x^4 - 14x^3 + 21x^2[/tex] factored by its greatest common factor is [tex]7x^2(x^2 - 2x + 3)[/tex].
