Answer :
We start with the polynomial
[tex]$$6x^3 - 10x^2 + 15x - 25.$$[/tex]
Step 1. Group the terms into two pairs.
Group the first two terms and the last two terms:
[tex]$$\bigl(6x^3 - 10x^2\bigr) + \bigl(15x - 25\bigr).$$[/tex]
Step 2. Factor out the greatest common factor (GCF) from each group.
For the first group, [tex]$6x^3 - 10x^2$[/tex], the GCF is [tex]$2x^2$[/tex]. Factoring gives:
[tex]$$6x^3 - 10x^2 = 2x^2(3x - 5).$$[/tex]
For the second group, [tex]$15x - 25$[/tex], the GCF is [tex]$5$[/tex]. Factoring gives:
[tex]$$15x - 25 = 5(3x - 5).$$[/tex]
Now the expression looks like:
[tex]$$2x^2(3x - 5) + 5(3x - 5).$$[/tex]
Step 3. Factor out the common binomial factor.
Both terms contain the common factor [tex]$(3x - 5)$[/tex], so factor it out:
[tex]$$2x^2(3x - 5) + 5(3x - 5) = (3x - 5)(2x^2 + 5).$$[/tex]
Step 4. Write down the final factorization.
The completely factored form of the polynomial is:
[tex]$$6x^3 - 10x^2 + 15x - 25 = (3x - 5)(2x^2 + 5).$$[/tex]
This is the final answer.
[tex]$$6x^3 - 10x^2 + 15x - 25.$$[/tex]
Step 1. Group the terms into two pairs.
Group the first two terms and the last two terms:
[tex]$$\bigl(6x^3 - 10x^2\bigr) + \bigl(15x - 25\bigr).$$[/tex]
Step 2. Factor out the greatest common factor (GCF) from each group.
For the first group, [tex]$6x^3 - 10x^2$[/tex], the GCF is [tex]$2x^2$[/tex]. Factoring gives:
[tex]$$6x^3 - 10x^2 = 2x^2(3x - 5).$$[/tex]
For the second group, [tex]$15x - 25$[/tex], the GCF is [tex]$5$[/tex]. Factoring gives:
[tex]$$15x - 25 = 5(3x - 5).$$[/tex]
Now the expression looks like:
[tex]$$2x^2(3x - 5) + 5(3x - 5).$$[/tex]
Step 3. Factor out the common binomial factor.
Both terms contain the common factor [tex]$(3x - 5)$[/tex], so factor it out:
[tex]$$2x^2(3x - 5) + 5(3x - 5) = (3x - 5)(2x^2 + 5).$$[/tex]
Step 4. Write down the final factorization.
The completely factored form of the polynomial is:
[tex]$$6x^3 - 10x^2 + 15x - 25 = (3x - 5)(2x^2 + 5).$$[/tex]
This is the final answer.
