In Exercises 45–68, factor by grouping.

x³ − 12 − 3x² + 4x

Question

In Exercises 45–68, factor by grouping.

x³ − 12 − 3x² + 4x

Accepted Answer

Hello, today we're going to be factoring the given polynomial. So what we are given is K cubed minus 35 minus seven K squared plus five K. Now, before we factor this given polynomial, let's go ahead and reorder the terms with respect to the exponents. So we're gonna want to write the terms in order from greatest exponents to least exponent. And that's going to allow us to rewrite the polynomial as K cubed minus seven K squared plus five K minus 35. So now that we've rewritten the polynomial noticed that we currently have a polynomial of four terms, having a polynomial of four terms is a clue that we should try factoring by grouping, factoring by grouping allows us to group up the first two terms together and the last two terms together. And what we want to do is factor out a common factor from each of the groups. So looking at the first group K cubed minus seven K squared has a common factor of K squared because in each of these terms, there is at least two instances of K. So if we factor out K squared from the first group, we can rewrite it as K squared times the quantity K minus seven. Now let's take a look at the second group. In the second group, we have five K minus 35 The coefficient five and the constant five are both multiples of five. So we can factor a five from these first two terms. And doing this is going to give us plus five times the quantity K -7. Now notice that we have two instances of K minus seven, we can go ahead and combine these into a single K minus seven by combining the leading coefficients together and combining the leading coefficients can be rewritten as K squared plus five times the quantity K minus seven. Now what we have is a sum of square terms and that is cannot be factored any further. So this is going to be the simplest form of the factored polynomial. And with that being said, the answer to this problem is going to be C so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 45–68, factor by grouping. x³ − 12 − 3x² + 4x

Key Concepts

Factoring by Grouping

Polynomial Terms

Common Factors

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