Question

which shows one way to determine the factors of $x^3 - 9x^2 + 5x - 45$ by grouping?
$\bigcirc\\ x^2(x - 9) - 5(x - 9)$
$\bigcirc\\ x^2(x + 9) - 5(x + 9)$
$\bigcirc\\ x(x^2 + 5) - 9(x^2 + 5)$
$\bigcirc\\ x(x^2 - 5) - 9(x^2 - 5)$

Explanation:

Step1: Group the terms

We have the polynomial \(x^{3}-9x^{2}+5x - 45\). Group the first two terms and the last two terms: \((x^{3}-9x^{2})+(5x - 45)\).

Step2: Factor out the GCF from each group

For the first group \(x^{3}-9x^{2}\), the greatest common factor (GCF) is \(x^{2}\), so we factor out \(x^{2}\) to get \(x^{2}(x - 9)\). For the second group \(5x-45\), the GCF is \(5\), but if we factor out \(- 5\) (to make the binomial inside the parentheses the same as in the first group), we get \(-5(-x + 9)=-5(9 - x)\) which is not helpful. Wait, actually, if we factor out \(5\) from \(5x - 45\), we get \(5(x - 9)\), but the sign is positive. Wait, no, let's re - group. Wait, another way: group as \(x^{3}+5x-9x^{2}-45=(x^{3}+5x)-(9x^{2}+45)\). Then factor out \(x\) from the first group: \(x(x^{2}+5)\) and factor out \(9\) from the second group: \(9(x^{2}+5)\), so we have \(x(x^{2}+5)-9(x^{2}+5)\). Let's check the first grouping method again. \((x^{3}-9x^{2})+(5x - 45)=x^{2}(x - 9)+5(x - 9)\)? Wait, no, the original options have \(-5\) in some. Wait, \(5x-45 = 5(x - 9)\), but in the options, the second group is factored with \(-5\). Wait, maybe I made a mistake in grouping. Let's look at the options. The third option is \(x(x^{2}+5)-9(x^{2}+5)\). Let's expand this: \(x\times x^{2}+x\times5-9\times x^{2}-9\times5=x^{3}+5x - 9x^{2}-45=x^{3}-9x^{2}+5x - 45\), which is the original polynomial. Let's check the first option: \(x^{2}(x - 9)-5(x - 9)=x^{3}-9x^{2}-5x + 45\), which is not the original polynomial. The second option: \(x^{2}(x + 9)-5(x + 9)=x^{3}+9x^{2}-5x - 45\), not the original. The fourth option: \(x(x^{2}-5)-9(x^{2}-5)=x^{3}-5x-9x^{2}+45=x^{3}-9x^{2}-5x + 45\), not the original. The third option: \(x(x^{2}+5)-9(x^{2}+5)=x^{3}+5x-9x^{2}-45=x^{3}-9x^{2}+5x - 45\), which matches the original polynomial.

Answer:

\(x(x^{2}+5)-9(x^{2}+5)\) (the third option: \(x(x^{2}+5)-9(x^{2}+5)\))