Question

Factor completely 2x^3 + 6x^2 + 10x + 30

Answer 1

The factored expression is 2(x² + 5)(x + 3).

Explanation:

Hey there!

We can use a factoring technique referred to as "grouping" to solve this problem.

Grouping is used for polynomials with four terms as a quick and easy factoring method to remove the GCF and get down to the initial terms that create the expression/function.

Grouping works in the following matter:

1. Given equation: ax³ + bx² + cx + d
2. Group a & b, c & d: (ax³ + bx²) + (cx + d)
3. Pull GCFs and factors

Let's apply these steps to the given equation.

1. Given equation: 2x³ + 6x² + 10x + 30
2. Group a & b, c & d: (2x³ + 6x²) + (10x + 30)
3. Pull GCFs and factors: 2x²(x + 3) + 10(x + 3)

As you'll see, we have a common term with both sides of the expression. This term, (x + 3), is a valuable asset to the factoring process. This is one of the factors for our expression.

Now, we use our GCFs to create another factor.

1. List GCFs: 2x², 10
2. Create a term: (2x² + 10)

Finally, we'll need to simplify this one by taking another GCF, 2.

1. Pull GCF: 2(x² + 5)

Now that we have this term, we need to understand that this could also be factored further using imaginary numbers, but it is also acceptable to leave it in this form.

Therefore, we have our final factors: 2(x² + 5) and (x + 3).

However, when we factor, we place all of our terms together. This leaves us with the final answer: 2(x² + 5)(x + 3).