Sure! Let's work through the given expression step-by-step.
Given expression:
[tex]\[ 2x^3y + 18xy - 10x^2y - 90y \][/tex]
### Part A: Factor out the greatest common factor (GCF)
1. Identify the GCF of all the terms:
- For the coefficients: The GCF of 2, 18, -10, and -90 is 2.
- For the variable parts: Each term contains the variable [tex]\(y\)[/tex] as a factor.
Thus, the GCF of the entire expression is [tex]\(2y\)[/tex].
2. Factoring out the GCF:
[tex]\[ 2x^3y + 18xy - 10x^2y - 90y = 2y(x^3 + 9x - 5x^2 - 45) \][/tex]
So after factoring out the GCF [tex]\(2y\)[/tex], the expression becomes:
[tex]\[ 2y(x^3 + 9x - 5x^2 - 45) \][/tex]
### Part B: Factor the entire expression completely
1. Rearrange the terms inside the parentheses in descending order of powers of [tex]\(x\)[/tex]:
[tex]\[ 2y(x^3 - 5x^2 + 9x - 45) \][/tex]
2. Grouping the terms to factor by grouping:
Group the terms in pairs that can help us factor further:
[tex]\[ 2y[(x^3 - 5x^2) + (9x - 45)] \][/tex]
3. Factor out the common factors in each group:
- From the first group [tex]\(x^3 - 5x^2\)[/tex], factor out [tex]\(x^2\)[/tex]:
[tex]\[ x^3 - 5x^2 = x^2(x - 5) \][/tex]
- From the second group [tex]\(9x - 45\)[/tex], factor out [tex]\(9\)[/tex]:
[tex]\[ 9x - 45 = 9(x - 5) \][/tex]
So, the expression becomes:
[tex]\[ 2y[x^2(x - 5) + 9(x - 5)] \][/tex]
4. Factor out the common binomial factor [tex]\((x - 5)\)[/tex]:
[tex]\[ 2y[(x - 5)(x^2 + 9)] \][/tex]
Thus, the completely factored form of the expression is:
[tex]\[ 2y(x - 5)(x^2 + 9) \][/tex]
To summarize the steps:
1. Original expression:
[tex]\[ 2x^3y + 18xy - 10x^2y - 90y \][/tex]
2. Factoring out the GCF [tex]\(2y\)[/tex]:
[tex]\[ 2y(x^3 + 9x - 5x^2 - 45) \][/tex]
3. Grouping terms:
[tex]\[ 2y[(x^3 - 5x^2) + (9x - 45)] \][/tex]
4. Factoring within groups:
[tex]\[ 2y[x^2(x - 5) + 9(x - 5)] \][/tex]
5. Factoring out the common binomial factor [tex]\((x - 5)\)[/tex]:
[tex]\[ 2y(x - 5)(x^2 + 9) \][/tex]
That's the complete solution for factoring the given expression step-by-step.
