- زمان مطالعه 7 دقیقه
- سطح خیلی سخت
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In the previous lesson, we talked about adding or subtracting algebraic expressions. In this lesson, we will begin the discussion of multiplying expressions. Looking at the two relatively simple cases, monomial times monomial, and monomial times binomial. Before we begin, let’s be clear on a few arithmetic principles.
It’s very important to realize when we get to algebra. All the algebra is obeying basic arithmetic principles. And so, we have to be clear in our arithmetic. Remember that multiplication is commutative and associative. This means that we can put factors in any order or groupings, and not change the product.
So for example, A times B times C times D. That’s exactly the same as B times A times D times C. We can change the order. We can group them. We can do A times D, and then B times C and then, multiply those two together. We could C by itself times BDA.
All of those are exactly the same. And there are many, many more, groupings and ordering’s possible. The point is, it doesn’t matter what order they’re in. It doesn’t matter how we group them. As long as everything’s getting multiplied together we will have the same product. That’s a really big idea about multiplication.
Also, we need to be clear on the Distributive Law. Multiplication distributes over addition and subtraction. So, if we multiply A times the quantity B plus C, we multiply each individual term. If we multiply A times the quantity B minus C, we multiply each individual term. Multiplication distributes over addition and subtraction. Multiplication does not distribute over multiplication.
So if 3 times x y does not equal 3 times x times 3 times y. We do not get two different factors of the three, one multiplying each factor. We don’t distribute over factors, we only distribute over sep, separate terms, and x y is a single term. So in fact, if we multiply 3 times x y, we just get 3 xy. Very important to be clear on this.
This is a subtle mistake. All right so now we can start talking multiplying. If we multiply a number a constant times a monomial with a variable the constant multiplies the coefficient. So for example, seven times five x squared, that’s just going to be 35 x squared.
We’re just going to multiply that constant times the coefficient. 2 times r to the fourth s squared t cubed, well, the coefficient is 1. So it will just be 2 times 1. It would just be 2 times that same combination of variables. What if we divide a monomial by a number? Remember that dividing by a number is the same as multiplying by its reciprocal.
That’s something we discussed in the fraction videos. So we have 15 x to the 6th, y to the 12th divided by 3. What happens is just that coefficient is gonna get divided. Now we can think of this as being multiply by one-third if we like. Just that coefficient is gonna be divided. The variables are gonna be unchanged.
And we’re gonna get 5 x of the 6, y to the 12. Now we will discuss multiplying two monomials, each of which contains variables. Remember that x times x is x squared. Something squared means something times itself.
And x squared times x is x cubed. For higher powers of x, recall the rule for multiplying powers. So when we multiply two powers x to the a, times x to the b, we simply add the powers. That’s x to the a plus b. So why this is true, and what’s going on with all these laws of exponents.
There are many other laws of exponents, and we’ll cover those in depth when we get to the Powers and Roots module. This is the only law of exponents that I’ll mention here, and this is probably familiar. When we multiply powers, we add the exponents. For example, suppose we multiply 3x times 4x squared.
Well first of all, we’re gonna multiply the coefficients, 3 times 4 is 12 and x times x squared is x cubed. So this product is going to be 12 x cubed. Now if we have this one, first of all mult, multiply the coefficients 7 times 6 is 42. The x squared time x will be x cubed.
Now the y square times y cubed. We have 2 factors of y times 3 factors of y that will give us 5 factors of y. We’ll add those coefficient, we’ll add the, the exponents of y, the 2 and the 3, and we’ll get, y to the 5th. And so the total product, will be 42, x cubed, y to the 5th. And the powers of the different variables stay separate.
It’s very important not to start mixing up the exponents of x with the exponents of y. You can get very confused if you don’t keep the variables separate. Here’s a practice problem. Pause the video and then we’ll talk about this. So these are the correct products here.
Now we’ll talk about the case of a monomial times a binomial. For this we use the Distributive Law. So A times B plus C equals, A times B plus A times C. This is proper distribution. So for example if we want to multiply 7x squared that monomial, times the binomial in parenthesis, we have to multiply that binomial times each term.
And then do the correct monomial times monomial multiplication for each term. And here we would get 7x to the 5th plus 14x to the 4th. Clearly, we could extend this pattern for a monomial times a trinomial or any higher polynomial. For example, here’s a monomial times a trinomial. We’ll multiply that 5x times each individual term.
And, we’ll wind up with this product. If there were more terms in the parentheses, we ould just distribute the monomial multiplication to each term. Here are some practice problems. Pause the video here and then we’ll talk about these. And here are the answers.
In summary, in this video, we’ve talked about the basics of algebraic multiplication. Including multiplying two monomials, and multiplying a monomial times a binomial or trinomial. In the next video, we will discuss the case of multiplying two binomials.
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