Q. 6.35TI - Intermediate Algebra

Question

Factor completely using trial and error:

$15n^3-85n^2+100n$

Step-by-Step Solution

Answer

The factors of $15n^3-85n^2+100n$ are $5n(3n-5)(n-4)$.

Step 1. Given and explanation

We have an equation $15n^3-85n^2+100n$. We will first take out the greatest common factor from the equation to reduce it to the simplest form.

Then we will find the factors accordingly and will get the factor set for the equation whose product will come out to be the given equation.

Step 2. Taking the greatest common factor out.

We have the equation and through it, we can see that the greatest common factor here is $5n$.

So after taking out the factor, we get a resulting equation as $5n(3n^2-17n+20)$.

Step 3. Finding out the factors.

We have $5n(3n^2-17n+20)$.

We will find factors of the first and late terms.

The first term is $3$ and it's factors are $(3\times 1)$.

The last term is $20$ and i's factors are$(1\times 20),(2\times 10),(4\times 5)$.

But we need to take care of the sign of the middle term as it is negative here. So the factors should both be positive or both be negative.

Step 4. Trying out and finding the correct factor set.

The possible factors are-

Possible factors	Product
$(3n-1)(n-20)$	$3n^2-61n+20$
$(3n-20)(n-1)$	$3n^2-23n+20$
$(3n-2)(n-10)$	$3n^2-32n+20$
$(3n-10)(n-2)$	$3n^2-16n+20$
$(3n-4)(n-5)$	$3n^2-19n+20$
$(3n-5)(n-4)$	$3n^2-17n+20$

Through this, we can see that the correct factor set is $(3n-5)(n-4)$.

Rewriting it with the greatest common factor out gives us $5n(3n-5)(n-4)$.

Step 5. Checking the solution

We will check the solution by multiplying. If we get the same given equation, our calculations are right.

So,

$$\begin{gathered} =5n(3n-5)(n-4) \\ =\left[15n^2-25n\right](n-4) \\ =15n^3-60n^2-25n^2+100n \\ =15n^3-85n^2+100n \end{gathered}$$

Thus our calculations are right.