What is one method to find the factors of the polynomial x³ + 5x² + 6x – 30 using grouping?

Finding the factors of the polynomial x³ + 5x² + 6x – 30 can be achieved through a method known as grouping. This technique involves rearranging and grouping terms in such a way that we can factor out common elements.

Step 1: Group the Terms

First, we will split the polynomial into two groups. We can group the first three terms together and separate the constant:

(x³ + 5x² + 6x) - 30

Step 2: Factor the First Group

Next, we will factor out the common factor from the first group:

x²(x + 5)

Now, the expression looks like this:

x²(x + 5) - 30

Step 3: Rearrange and Factor the Second Group

To make it easier to factor out the remaining part, we can adjust our approach to include the constant with a term from our first group. Let’s rewrite the polynomial in such a way that it makes the terms clearer:

x³ + 5x² + 6x - 30 = (x³ + 5x²) + (6x - 30)

This gives us two new groups:

=(x²(x + 5) + 6(x - 5))

Step 4: Factor Out Common Binomial

Now we can see that both groups share a common binomial factor:

(x + 5)(x² + 6)

Final Factors

So, we conclude that the polynomial x³ + 5x² + 6x – 30 can be factored as:

(x + 5)(x² + 6)

We can then verify by expanding:

(x + 5)(x² + 6) = x³ + 6x + 5x² + 30 = x³ + 5x² + 6x - 30

Thus, the polynomial is factored correctly!