In Exercises 1–22, factor the greatest common factor from each po...

Question

In Exercises 1–22, factor the greatest common factor from each polynomial.

12x⁴ − 8x²

Accepted Answer

Welcome back. I am so glad you're here. We are asked to factor the following polynomial were given P to the fourth minus six P squared. Answer choice A is P squared times parentheses, 14 P squared minus six and parentheses answer choice B is two P squared times parentheses, seven P squared minus three and parentheses, answer choice C is two P squared times parentheses, seven P squared minus three P and parentheses and answer choice D is two P times parentheses, seven P squared minus three P and parentheses. Well, we recall from previous lessons that the first step in factoring polynomial is we're going to compare the parts of each term that are similar. So let's start with the coefficients. We have 14 and negative six and we ask ourselves between 14 and negative six. Do they share any common factors? And if so what's the greatest common factor? And the greatest common factor between 14 and negative six is two. So they are both divisible by two, we can factor out a two. Now, the next part that the parts that are alike are these peas. So we can compare e to the fourth and P squared. While we recall from previous lessons that when we are looking to factor variables, we look for the variable that has the lowest degree. So between the degrees four and two, which one is the lowest, that's too. So we're going to divide both terms by P squared or in other words, we are factoring out P squared now that we know what we've factored out, we need to find out what's left behind. So we're taking 14 P to the fourth divided by two P squared. Let's start with the coefficients 14, divided by two. That is seven. So there's seven here inside the parentheses. And now P to the fourth divided by P squared. Well, we recall from previous lessons that when we're dividing variables, we're going to subtract their exponents. So we're taking four minus 24 minus two is two. So we have P squared left in the first term. We've divided the first term we factored it. Now let's do the second term negative six P squared divided by two P squared, negative six, divided by two equals a negative three. So we can put minus three and P squared divided by P squared. Well, that's two minus two, which is zero. We have P two, the zero, we know that anything raised to the zero power equals one. So we're taking that negative three times one. Well, negative three times one is negative three. So we don't need to put that times one and we can just close the parentheses. This polynomial is as factored as it's going to get. So now we look at our answer choices for two P squared times parentheses, seven P squared minus three. And that matches with answer choice B while done, we'll catch you on the next one.

In Exercises 1–22, factor the greatest common factor from each polynomial. 12x⁴ − 8x²

Key Concepts

Greatest Common Factor (GCF)

Factoring Polynomials

Polynomial Terms

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