WEBVTT mathematics/algebra-2/eaton
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Welcome to Educator.com.
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Today, we are going to be working on operations on polynomials.
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And for this lesson, we are going to cover addition, subtraction, and multiplication.
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Division is covered separately under another lesson.
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OK, reviewing how to add and subtract polynomials: add or subtract by removing the parentheses and combining like terms.
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And recall that like terms, or like monomials, have the same variables to the same powers.
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So, like terms have the same variables raised to the same powers.
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For example, 2x and 6x are like terms; y⁴ and 5y⁴ are like terms; or x²z⁵ and 8x²z⁵ are like terms.
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They have the same variables raised to the same powers, and they can be combined.
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So, talking about adding polynomials; what you need to do is add the like terms together.
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These are monomials; and now we are working with polynomials.
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For example, if I have 5x² + xy + 4z, and I need to add that to 6x² + 2xy - 6z,
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for addition, you just simply move the parentheses; the signs within the parentheses stay the same.
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So, remove the parentheses, and combine like terms: I have 5x² and 6x², so that is going to give me 11x².
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xy and 2xy gives me 3xy; 4z - 6z is -2z; that is addition.
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Subtraction--you just have to be careful with the signs.
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For subtraction: 3x² + 2xy + 5y, for example, minus 2x³...let's see...- 4x + 6y.
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OK, now, for this first one, I can just remove the parentheses, because there is no negative sign in front of this, so I just take the parentheses away.
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Now, when there is a negative sign, in order to remove the parentheses, make this addition;
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and we are going to apply the negative sign to each term within the parentheses.
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So, 2x³ becomes -2x³; -4x...if I take a negative and a negative, I will get + 4x.
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If I take the negative and apply it to 6y, that is going to give me -6y.
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Now, that allowed me to remove the parentheses; but I had to reverse the signs within the parentheses where the negative sign was applied.
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OK, now, combining like terms: if I have 3x³ - 2x³, that is going to give me just x³.
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2x + 4x is 6x; 5y - 6y is -y; so again, just be careful when you are removing the parentheses,
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when there is a negative sign in front; you have to apply that negative sign to each term within the parentheses.
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OK, when multiplying polynomials, you need to use the distributive property.
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Recall that the distributive property states that a times (b + c) equals ab + ac.
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There can be more than two terms in here, and the distributive property would apply in that case, as well, to give you ab + ac + ad, and so on.
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So, let's look at multiplying a monomial by a polynomial, for example.
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If I have 3x² times (2x³ + x - 5), I am going to use the distributive property right here;
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and I am going to multiply 3x² times 2x³, + 3x² (now I am going
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to apply that to this term) times x, and then 3x² again, times -5.
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And this is going to give me 6x...and then, since I am going to be multiplying, I need to add the exponents.
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So, these have the same base of x, but I need to add the exponents; that is going to give me 2 + 3; that is going to be x⁵.
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OK, now this is going to be 3x² times x, so this is 2 + (this really is a) 1, so that is x³.
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And then, 3 times -5 is -15x²; and then, I am writing this out as 6x⁵ + 3x³,
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and then I am just going to make this - 15x².
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And I can't go any further, because I cannot combine terms, since these are the same variable, but they are raised to different powers.
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So, I can't simply combine those.
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Looking at the first example, this is simply adding a polynomial to another polynomial.
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And I am going to remove the parentheses; and since it is addition, I don't have to worry about any sign changes.
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I just take away the parentheses and keep the signs the same.
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So, remove the parentheses, and then combine like terms.
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This is going to give me -3x² + 8x², is going to be 5x².
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-4x and -3 is -7x; 7 minus 8 is -1.
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So, this is very straightforward: just remove the parentheses and combine like terms.
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Example 2 is asking me to simplify; and this time, it is subtraction, so I have to be a little more careful with removing the parentheses.
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The first one does not have a negative sign in front of it, so I simply take out the parentheses.
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Here, I am going to change this to a plus; but in order to do that, I have to apply the negative sign to each term within the parentheses.
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A negative and a negative is going to give me positive 2x³.
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A negative and a negative here is going to give me + 8y.
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And then, a negative and a positive here--that is just going to be -y².
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Now, I have gotten rid of the parentheses; I can combine like terms.
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I have only one y³, so that is going to just stay by itself; this is 9y³.
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Let's look for y² terms: well, I have -3y², and I have -y²; that is going to be -4y².
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That is my y² terms; now I have a constant right here, positive 7.
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And I am looking, and I don't see any other constants; so that is just going to stay by itself.
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And then, over here, I actually have an x³ term, so this is going to also remain as it is, because there is nothing I can combine it with.
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So, this is going to give me 9y³ - 4y² + 2x³...let's put the constant at the end here.
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So again, the important point here is that, when you have a negative sign,
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and you are applying it, you need to make sure that you reverse the sign for each term inside the parentheses.
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OK, Example 3: we are multiplying a monomial by a polynomial, and we are going to use the distributive property.
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That states that a times (b + c) equals ab + ac, and we can apply that to a third or additional terms, as well, inside the parentheses.
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So, this is going to give me 4a²b times -5a³b²,
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plus...again, I am multiplying 4a²b times the next term, 6a²b²,
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and then, times the third term, 4a²b times -4ab³.
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OK, so I am multiplying each of these out--multiplying 4 times -5 gives me -20; recall that, if exponents have the same bases,
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I am going to multiply by adding the exponents (this rule).
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It is a^m times a^n equals a^m + n.
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So, this is going to give me 2 + 3, so that is a⁵; b...this is really a 1 implied, and 1 and 2 is 3: b³.
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Here, I have 6 and 4, so that is 24; a² times a²...2 + 2 would be a⁴.
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b to the first power, times b to the second...1 + 2 gives me b raised to the third power.
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4 and -4 is -16; a² times a is really 2 + 1, so that is going to give me a³.
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And then, b (really to the first power) times b³ is b⁴.
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Now, I am looking to see if I have any like terms; and I don't.
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There are no terms here where it is the same bases to the same power; if there were, I would combine them to complete my simplification.
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But none of these are like terms, so I am done simplifying this expression.
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OK, again, I am asked to simplify; and this time, I am multiplying a monomial times a binomial times a binomial, so it is a little more complicated.
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And you could go about it by multiplying this...applying it to each term first;
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or what I am going to actually do is multiply the binomials out and then work with this.
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So, I need to apply the distributive property more than once here.
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I am going to start out with this, keeping the 4z², and now multiplying out using the distributive property.
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And recall: with two binomials, you can use the FOIL method.
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So, multiplying the first terms is going to give me 15z².
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Now, the outer terms--that is 12z; the inner terms: -35z; and finally, the last terms, multiplied by each other, are -28.
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I can simplify a bit more, and this is 15z²; I have like terms in here, and 12 - 35 is -23z, minus 28.
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OK, so I applied the distributive property in here, multiplying each term in one times each term inside the other parentheses.
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Now, I need to apply the distributive property, again, to work with the 4z².
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And that is going to give me 4z² times 15b² plus 4z² times -23z, plus 4z² times -28.
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OK, so I get 4 times 15; that is 60; remember that a^m times a^n equals a...and then I add the exponents.
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So, I am adding 2 and 2 to get 4; this is z⁴.
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-23 times 4 is -92; z² times z to the first power is...2 + 1 is 3.
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Now, 4 times -28 is -112; and here, I just have a z²--no z term there.
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OK, so I am looking here to see if I can simplify further, and I cannot, because there are no like terms.
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So, today we covered operations on polynomials, focusing on addition, subtraction, and multiplication.
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And next time, we will talk about division with polynomials.
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