To factor the polynomial [tex]\( 14x^5 - 28x^3 + 7x^2 \)[/tex] completely, let's follow a detailed, step-by-step process:
1. Identify the Greatest Common Factor (GCF) of the Coefficients:
- The coefficients of the terms are 14, 28, and 7.
- To find the GCF of 14, 28, and 7:
- The prime factorization of 14 is [tex]\( 2 \times 7 \)[/tex].
- The prime factorization of 28 is [tex]\( 2^2 \times 7 \)[/tex].
- The prime factorization of 7 is [tex]\( 7 \)[/tex].
- The highest common factor among these coefficients is [tex]\( 7 \)[/tex].
2. Identify the Common Variable Factor:
- The variable terms are [tex]\( x^5 \)[/tex], [tex]\( x^3 \)[/tex], and [tex]\( x^2 \)[/tex].
- The lowest power of [tex]\( x \)[/tex] among these terms is [tex]\( x^2 \)[/tex].
- Thus, the common variable factor is [tex]\( x^2 \)[/tex].
3. Combine the GCF of the Coefficients and the Common Variable Factor:
- The GCF of the polynomial [tex]\( 14x^5 - 28x^3 + 7x^2 \)[/tex] is [tex]\( 7x^2 \)[/tex].
4. Factor out the GCF from each term of the polynomial:
- When we factor [tex]\( 7x^2 \)[/tex] out from [tex]\( 14x^5 \)[/tex]:
[tex]\[
14x^5 \div 7x^2 = 2x^3
\][/tex]
- When we factor [tex]\( 7x^2 \)[/tex] out from [tex]\( 28x^3 \)[/tex]:
[tex]\[
28x^3 \div 7x^2 = 4x
\][/tex]
- When we factor [tex]\( 7x^2 \)[/tex] out from [tex]\( 7x^2 \)[/tex]:
[tex]\[
7x^2 \div 7x^2 = 1
\][/tex]
5. Rewrite the Polynomial with the Factored GCF:
- The polynomial [tex]\( 14x^5 - 28x^3 + 7x^2 \)[/tex] can thus be rewritten as:
[tex]\[
14x^5 - 28x^3 + 7x^2 = 7x^2 (2x^3 - 4x + 1)
\][/tex]
Therefore, the factored form of [tex]\( 14x^5 - 28x^3 + 7x^2 \)[/tex] is:
[tex]\[
7x^2 (2x^3 - 4x + 1)
\][/tex]
