To determine the factors of the polynomial [tex]\(x^3 - 9x^2 + 5x - 45\)[/tex] by grouping, let's look at how we can rearrange and group the terms to factor it effectively.
Given polynomial:
[tex]\[ x^3 - 9x^2 + 5x - 45 \][/tex]
We need to group the terms in pairs and factor out the common factor in each group. Let's start by grouping the terms:
[tex]\[ (x^3 - 9x^2) + (5x - 45) \][/tex]
Next, we factor out the greatest common factor (GCF) from each group:
[tex]\[ x^2(x - 9) + 5(x - 9) \][/tex]
Now, notice that each group contains a common binomial factor [tex]\( (x - 9) \)[/tex]. We can factor this common binomial factor out:
[tex]\[ (x - 9)(x^2 + 5) \][/tex]
Thus, we have factored the polynomial [tex]\(x^3 - 9x^2 + 5x - 45\)[/tex] as:
[tex]\[ (x - 9)(x^2 + 5) \][/tex]
Looking at the given choices:
1. [tex]\( x^2(x - 9) - 5(x - 9) \)[/tex]
2. [tex]\( x^2(x + 9) - 5(x + 9) \)[/tex]
3. [tex]\( x(x^2 + 5) - 9(x^2 + 5) \)[/tex]
4. [tex]\( x(x^2 - 5) - 9(x^2 - 5) \)[/tex]
The first option matches the grouping we derived:
[tex]\[ x^2(x - 9) - 5(x - 9) \][/tex]
So, the correct way to demonstrate one method of factoring [tex]\(x^3 - 9x^2 + 5x - 45\)[/tex] by grouping is:
[tex]\[ x^2(x - 9) - 5(x - 9) \][/tex]
Which shows the final factored form as:
[tex]\[ (x - 9)(x^2 + 5) \][/tex]
Therefore, the correct choice is:
[tex]\[ x^2(x - 9) - 5(x - 9) \][/tex]
