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Lesson 117: Working with Polynomials (Part 2 of 2)
This is a continuation of Lesson 116. Make sure that you study that lesson first.Multiplying Polynomials
There are several combinations of multiplication that come into play. First is multiplying a monomial times a polynomial, for example, 4x(3x2 - 6x - 7). What we do is use the distributive property, and multiply each term of the polynomial by 4x. Again, we must be careful with our signs. We end up with 12x3 - 24x2 - 28x. Remember how we multiply terms with a common base. We add the exponents. That's how (4x)(3x2) became 12x3.
Now let's try a binomial times a binomial, for example, (x + 2)(x + 3). What we do is use what is called the FOIL method. First we'll multiply the first terms of each binomial. Then we'll multiply the outer terms. Then the inner terms, then the last terms. We'll combine our result as is possible.
Let's write out some binomials using letters, so it is easier to see what we're doing: Let's say we have (x + a)(x + b). Our answer will be x2 (first) + bx (outer) + ax (inner) + ab (last). Then we'll combine like terms where possible.
Let's get back to our example above. We have x2 + 3x + 2x + 6. We'll combine the x terms, and end up with x2 + 5x + 6.
Let's try another one: (x + 2)(2x - 5). Using FOIL, we get 2x2 - 5x + 4x - 10. Make sure that you fully see how the signs work. The -5 in the second binomial is really negative 5. Remember that a negative times a positive is a negative. Take these problems slow, and be careful with your signs. Make sure you're observing all of the math rules that we've learned up to this point. Getting back to our problem, we need to combine like terms. The -5x + 4x equal -1x, or just -x. Our answer is 2x2 - x - 10.
Dividing Polynomials
First we'll learn how to divide a polynomial by a monomial.
For example:
(16x2 - 10x + 6) / 2. I've used parentheses so that
it is easy to see what is in the numerator and what is in the
denominator. Remember, an expression in fraction format like
this is really just a division problem. All we do is divide
each term in the polynomial by the monomial. We get 8x2
- 5x + 3.
Let's try another: (20x4 - 15x3 + 5x2) / 5x. We have to remember our rule for dividing exponents. We actually subtract them. We get 4x3 - 3x2 + x. Make sure you understand how we got that. Remember that when we do 5/5, we get 1, which we just drop.
Sometimes division problems can be very tricky. Here's
quite a tricky one:
(8x5 + 6x + 7) / 2x3. We'll divide each
term by the monomial. We'll do 8x5 / 2x3
and get 4x2. Now we'll do 6x / 2x3.
Remember that we have to subtract our exponents, and that the x
really has an exponent of 1. We get 3x-2. Now
we'll try to do 7 / 2x3, but 2 does not divide evenly
into 7, so we just have to leave it.
What have is 4x2 + 3x-2 + (7 / 2x3). Usually we do not like to leave negative exponents. Remember the rule that we can make the exponent positive, and put it in the denominator. That means we'll have: 4x2 + (3 / x2) + (7 / 2x3). As messy as that looks, we have to leave our answer like that. There is nothing else that we can combine.
Undefined Fractions
The final thing to talk about is undefined fractions There is a rule in math which says that we are not allowed to divide by 0. Stated another way, the denominator can never be 0, but it's OK if the numerator is. Make sure that you memorize that rule. Later you'll learn why that rule makes sense. A fraction in which you divide by 0 is considered to be undefined. For example, you might be asked what values of x will make this fraction undefined: (17x2 + 3x + 5) / (x - 3). Remember, we're only concerned about the denominator. You can see that when x equals 3, the denominator will equal 0, so that means that x = 3 will make this fraction undefined.
You'll have much more practice with this in later lessons. Make sure that you fully understand everything that was presented in this lesson.
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