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Answer :
We are given the polynomial
[tex]$$
25 x^7 - 10 x^5 + 15 x^3 - 5 x^2.
$$[/tex]
Step 1. Factor out the greatest common factor (GCF).
Each term in the polynomial has factors of 5 and at least [tex]$x^2$[/tex]. We factor out [tex]$5x^2$[/tex]:
[tex]$$
25x^7 = 5x^2 \cdot 5x^5,\quad -10x^5 = 5x^2 \cdot (-2x^3),\quad 15x^3 = 5x^2 \cdot 3x,\quad -5x^2 = 5x^2 \cdot (-1).
$$[/tex]
Thus, factoring out the GCF gives:
[tex]$$
25 x^7 - 10 x^5 + 15 x^3 - 5 x^2 = 5x^2 \left(5x^5 - 2x^3 + 3x - 1\right).
$$[/tex]
Step 2. Write the fully factorized form.
By factoring out [tex]$5x^2$[/tex], the polynomial is expressed as:
[tex]$$
5x^2 \left(5x^5 - 2x^3 + 3x - 1\right).
$$[/tex]
At this point, the polynomial is completely factored as no further common factors can be extracted from the inner polynomial.
Final Answer:
[tex]$$
5x^2 \left(5x^5 - 2x^3 + 3x - 1\right).
$$[/tex]
[tex]$$
25 x^7 - 10 x^5 + 15 x^3 - 5 x^2.
$$[/tex]
Step 1. Factor out the greatest common factor (GCF).
Each term in the polynomial has factors of 5 and at least [tex]$x^2$[/tex]. We factor out [tex]$5x^2$[/tex]:
[tex]$$
25x^7 = 5x^2 \cdot 5x^5,\quad -10x^5 = 5x^2 \cdot (-2x^3),\quad 15x^3 = 5x^2 \cdot 3x,\quad -5x^2 = 5x^2 \cdot (-1).
$$[/tex]
Thus, factoring out the GCF gives:
[tex]$$
25 x^7 - 10 x^5 + 15 x^3 - 5 x^2 = 5x^2 \left(5x^5 - 2x^3 + 3x - 1\right).
$$[/tex]
Step 2. Write the fully factorized form.
By factoring out [tex]$5x^2$[/tex], the polynomial is expressed as:
[tex]$$
5x^2 \left(5x^5 - 2x^3 + 3x - 1\right).
$$[/tex]
At this point, the polynomial is completely factored as no further common factors can be extracted from the inner polynomial.
Final Answer:
[tex]$$
5x^2 \left(5x^5 - 2x^3 + 3x - 1\right).
$$[/tex]
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