Multiplying Expressions
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 we get to algebra, all of the algebra is a bang basic arithmetic principles. And so we have to be clear on our arithmetic. Remember that multiplication is commutative and associative. This means that we can put back these in any order or groupings and not change the product.
So for example, A x B x C x D, that's exactly the same as B x A x D x C, we can change the order. We can group them, we can do A x D and then B x C and then multiply those two together. We could do C by itself times BDA. All of those are exactly the same and there are many, many more groupings and orderings possible.
The point, is it does not matter what order they are in, it does not matter how we group them as long as everything is getting multiplied together, we will have the same product. That's really a big idea about multiplication. Also, we need to be clear on the Distributive Law. Multiplication distributes over addition and subtraction.
So if we multiple A times the quantity B + C, we multiple each individual term. If we multiply A times the quantity B- C, we multiply each individual term. Multiplication distributes over addition and subtraction. Multiplication does not distribute over multiplication. So 3 x (xy) does not equal (3x)(3y). We do not get two different factors of the three, one multiplying each factor.
We don't distribute over factors, we only distribute over separate terms and xy is the single term. So the fact that we multiply 3 times xy we just get 3xy. Very important to be clear on this, this is a subtle mistake. All right, so now we can start talking about multiplying. If we multiply a number, a constant, times a monomial with a variable, the constant multiplies the coefficient.
So for example, 7 x (5x squared), that's just gonna be 35x squared. We're just gonna multiply that constant times the coefficient. Two times r to the fourth, s squared t cubed. Well, the coefficient is one, so it would just be 2 x 1, it would just be two 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 discuss in the fraction videos. So we have 15 x to the 6th, y to the 12th by 3, what happens is just that coefficient is gonna get divided. Now we can think of this as being multiplied by 1/3, if we like. Just that coefficient is gonna divided, the variables are gonna be unchanged and we're going to get 5x to the 6th y to the 12th.
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, (xa)(xb), 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) (4X squared). Well, first of all, we're going to multiply the coefficients. Three times four is 12 and x times x squared, is x cubed. So this product is going to be 12x cubed. Now if we have this one.
First of all, we'll multiply the coefficients. Seven times six is 42. The x squared times x, will be x cubed. Now the y squared times y cubed, we have two factors of y times three factors of y, that would give us five factors of y. We'll add this coefficient where at the exponents of y, the 2 and the 3 and we'll get y to the 5th.
And so the total product will be 42x 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 are some practice problems.
Pause the video and then we'll talk about these. So these are the correct products here. Now we'll talk about the case of a minomial times a binomial. For this we use the distributive law. So A(B+C) = AB + AC. This is proper distribution.
So, for example, if we wanna multiply 7x squared, that monomial, times the binomial in parentheses, 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 + 14x to the 4th. Clearly, we could extend this patter for a monomial times a trinomial or any higher polynomial.
For examples, 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 would just distribute the monomial multiplication to each term. Here some practice problems, pause the video here, and then we'll talk about these.
And here the answers. In summary, in this video, we 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.
Read full transcriptIt's very important to realize when we we get to algebra, all of the algebra is a bang basic arithmetic principles. And so we have to be clear on our arithmetic. Remember that multiplication is commutative and associative. This means that we can put back these in any order or groupings and not change the product.
So for example, A x B x C x D, that's exactly the same as B x A x D x C, we can change the order. We can group them, we can do A x D and then B x C and then multiply those two together. We could do C by itself times BDA. All of those are exactly the same and there are many, many more groupings and orderings possible.
The point, is it does not matter what order they are in, it does not matter how we group them as long as everything is getting multiplied together, we will have the same product. That's really a big idea about multiplication. Also, we need to be clear on the Distributive Law. Multiplication distributes over addition and subtraction.
So if we multiple A times the quantity B + C, we multiple each individual term. If we multiply A times the quantity B- C, we multiply each individual term. Multiplication distributes over addition and subtraction. Multiplication does not distribute over multiplication. So 3 x (xy) does not equal (3x)(3y). We do not get two different factors of the three, one multiplying each factor.
We don't distribute over factors, we only distribute over separate terms and xy is the single term. So the fact that we multiply 3 times xy we just get 3xy. Very important to be clear on this, this is a subtle mistake. All right, so now we can start talking about multiplying. If we multiply a number, a constant, times a monomial with a variable, the constant multiplies the coefficient.
So for example, 7 x (5x squared), that's just gonna be 35x squared. We're just gonna multiply that constant times the coefficient. Two times r to the fourth, s squared t cubed. Well, the coefficient is one, so it would just be 2 x 1, it would just be two 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 discuss in the fraction videos. So we have 15 x to the 6th, y to the 12th by 3, what happens is just that coefficient is gonna get divided. Now we can think of this as being multiplied by 1/3, if we like. Just that coefficient is gonna divided, the variables are gonna be unchanged and we're going to get 5x to the 6th y to the 12th.
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, (xa)(xb), 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) (4X squared). Well, first of all, we're going to multiply the coefficients. Three times four is 12 and x times x squared, is x cubed. So this product is going to be 12x cubed. Now if we have this one.
First of all, we'll multiply the coefficients. Seven times six is 42. The x squared times x, will be x cubed. Now the y squared times y cubed, we have two factors of y times three factors of y, that would give us five factors of y. We'll add this coefficient where at the exponents of y, the 2 and the 3 and we'll get y to the 5th.
And so the total product will be 42x 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 are some practice problems.
Pause the video and then we'll talk about these. So these are the correct products here. Now we'll talk about the case of a minomial times a binomial. For this we use the distributive law. So A(B+C) = AB + AC. This is proper distribution.
So, for example, if we wanna multiply 7x squared, that monomial, times the binomial in parentheses, 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 + 14x to the 4th. Clearly, we could extend this patter for a monomial times a trinomial or any higher polynomial.
For examples, 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 would just distribute the monomial multiplication to each term. Here some practice problems, pause the video here, and then we'll talk about these.
And here the answers. In summary, in this video, we 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.
