Factoring by Grouping

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En provenance du cours de University of California, Irvine

Pre-Calculus: Functions

347 notes

University of California, Irvine

Pre-Calculus: Functions

347 notes

This course covers mathematical topics in college algebra, with an emphasis on functions. The course is designed to help prepare students to enroll for a first semester course in single variable calculus. Upon completing this course, you will be able to: 1. Solve linear and quadratic equations 2. Solve some classes of rational and radical equations 3. Graph polynomial, rational, piece-wise, exponential and logarithmic functions 4. Find integer roots of polynomial equations 5. Solve exponential and logarithm equations 6. Understand the inverse relations between exponential and logarithm equations 7. Compute values of exponential and logarithm expressions using basic properties

À partir de la leçon

Quadratic Equations.

In this module, we explore a variety of methods for solving quadratic equations.

Introduction to Polynomials and Rational Functions7:56

Factoring a Quadratic Expression10:22

Special Factoring Formulas 9:46

Factoring by Grouping 7:11

Solving a Quadratic Equation 2:44

Rencontrer les enseignants

Dr. Sarah Eichhorn
Associate Dean, Distance Learning & Tenured Lecturer
Dr. Rachel Cohen Lehman

[SOUND] Let's look at factoring by grouping.

[SOUND] For example, let's factor the following expression by grouping.

Now, to factor means we want to write this expression as a product of other

numbers or algebraic expressions. Now looking here, there is nothing in

common to all four of these terms, and when that's the case, we try to group

terms together and look for common factors.

And we can begin by trying to group the first two terms together and the last two

terms together. In other words, this is equal to

(4v^5+v^4) and then plus (20v+5). And now the greatest common factor in the

first two terms is v^4, factoring that out, we're left with

(4v+1). And the greatest common factor in the

second two terms, is a 5, factoring that out, we're left with

(4v+1), which is the same binomial expression in

the first grouping. Which is why factoring by grouping works

here, because now, we can factor that out of both of these.

In other words, this is equal to (4v+1*v^4+5),

which would be our answer. Let's look at another example.

[SOUND] Again, let's factor this expression by grouping and we'll start in

the same way, we'll group the first two terms together

as well as the last two terms. In other words, this is equal to

(5v^3-4w^2)+(-25w+20). And now, the greatest common factor in

the first two terms is a w^2, and when we factor that out, we're left

with (5w-4). What is the greatest common factor in this second grouping? Well, we

get factor either a five out or a negative five out.

But remember, our hope is that we're going to get the same binomial leftover.

So what happens when we factor out a five here? We'd be left with (-5w+4), which is

not that same binomial. However, if we factor out a negative 5,

we're left with (5w-4), which is that same binomial.

So that's what we want to factor out is the negative five,

so this is minus 5(5w-4). And now, we can factor this binomial out

of each of these groupings, which gives us (5w-4)(w^2-5), which would be our

answer. Alright. Let's see one more example.

[SOUND] Again, let's factor this by grouping and we'll begin in the same way,

we'll group together the first two terms and then the last two terms, which gives

us. (3pr-qr) and then plus (6ps-2qs).

Now, the greatest common factor in these first two terms is an r, when we factor

that out, we're left (3p-q). Now remember, our hope is is to get the

same binomial leftover in the second grouping.

And if we pull out a 2s, we'll be left with that same binomial, won't we?

(3p-q). So let's do that.

So we have plus 2s(3p-q), and then factoring that out of each of these

groupings, gives us (3p-q)*(r+2s),

which would be our answer. Now, in all these examples, we've been

starting with grouping the first two terms and then the last two terms.

But what if we started a little bit different here? So we still have the six

expression, 3pr-qr+6ps-2qs, but instead of grouping the first two terms and the

last two terms, what if we group the first term and third term together and

then the second term and the fourth term? In other words, let's write this as

(3pr+6ps)+(-qr-2qs). Now, the greatest common factor in the

first two terms is a 3p, so let's pull that out,

and we're left with (r+2s). Now remember, in the second grouping over

here, our hope is that after factoring we have the same binomial leftover.

So if we factor out a -q here, we'll be left with (r+2s),

which is the same binomial. So let's do that.

So we have -q(r+2s). And now we can factor out that binomial,

which leaves us with (r+2s)(3p-q). And by commutativity, that's the same as

this, isn't it? Okay. And this is how we factor by grouping.

Thank you and we'll see you next time. [SOUND]

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