Hi! Your answer is x - 5
Please see an explanation for a better and clear understanding to your problem!
Any questions about my answer and explanation can be asked through comments! :)
Step-by-step explanation:
To find an expression that is a factor, we simply factor the expression first. Sound a bit confusing? Well It's not that confusing if you understand it.
So what we need to do is to factor the expression first.
[tex]\huge{x^3-2x^2-15x}[/tex]
Notice anything? Yes, the whole expression has x-term as a variable. We can simply factor the expression by common factor, pulling the variable with the least degree.
We should get [tex]\large{x(x^2-2x-15)}[/tex] from common factor. However, we can still factor inside the brackets. Inside the bracket, the expression can be factored by two brackets factor. Our x-term which is outside of bracket is a bit useless so you can ignore it. What we want is to factor the expression inside the bracket.
[tex]\large{x^2-2x-15}[/tex]
To factor this kind of expression, we use two brackets factoring for a = 1.
[tex]\large{(x-a)(x-b)}[/tex]
To find the first term, we simply multiply x and x to each others.
To find the second term, we add -bx and -ax to each others.
To find the first term, we multiply -a and -b to each others.
So if we follow these steps, our (x-a)(x-b) should be x²-ax-bx+ab = x²-(a+b)x+ab
[tex]\large{x^2-(a+b)x+ab}[/tex]
We only need to find a and b value. Which two numbers multiplying each others and get -15? There are only -5 and 3 or 3 and -5.
That means -5 and 3 | 3 and -5 are our only options.
But what two numbers adding up each others and get -2? That means it's -5 and 3 because -5+3 = -2.
Therefore, from x²-2x-15. Our factors are (x-5)(x+3)
So when the expression is completely factored, the new factored expression is x(x-5)(x+3) Then we look at the choices to see if there are these factors in the choices. From the choices and the expression, the only correct answer is (x-5) because x+5 is not a factor (it doesn't appear), same with other two choices. They seem to be factored but not fully factored.
Hence, the answer is x - 5
