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In this video, we will learn about adding and subtracting polynomials. A polynomial is a type of
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algebraic expression. Specifically, a polynomial in x is a finite sum of terms in the form a x to
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the n where a is a real number and n is a whole number. This is an example of a polynomial. This
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polynomial has four terms. Each of these terms can be written in the form a x to the n. A polynomial
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whose powers are written in descending order as this one is is known to be in standard form.
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The numerical coefficient in each of these terms is known as the coefficient.
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For the first term x to the third, the coefficient is implied. There is a one as the numerical factor in front of x cubed.
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So the coefficient is one. For this term negative two x squared, the coefficient is negative two. For the term three x,
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the coefficient is three. And for the term five, notice that five is a coefficient, but
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because it doesn't have a variable in this term, we call this the constant term or just constant.
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An example of an expression that is not a polynomial would be x to the negative two
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plus eight x minus nine, and the reason this is not a polynomial is because the exponent
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is not a whole number. Each term of a polynomial has a degree. The degree of a term is the sum of
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the exponents on the variables contained in the term. The degree of a polynomial is the
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greatest degree of any term of the polynomial. Let's take a look at a couple of examples.
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Determine if the expression is a polynomial, and if it is a polynomial, state the degree.
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Our first expression is x squared minus eight x to the fifth plus six x to the third plus nine x
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minus five. To determine whether this expression is a polynomial, we look at the exponents on the
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variables. We notice that these are whole numbers, and so this is a polynomial.
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Next, to find the degree we look for the degree of each term, and we identify what the highest
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degree of the terms are. This degree is two. The degree of this term is five.
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This is three. The degree here is one. The degree here is zero, and that's because we can write this
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as negative five x to the zero power. The largest degree of all of these terms
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is five. So five is the degree of the polynomial. So yes, this is a polynomial of degree five.
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Notice that this polynomial is not written in standard form because the powers are not in descending order.
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The next example we have is x plus x to the one-half minus six. Let's look at the powers on the variables. We
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have one, one-half, and zero. Since one-half is not a whole number, this is not a polynomial.
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Remember polynomials can contain more than one variable as long as the
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exponents on the variables are whole numbers. In these examples, we'll determine the degree
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of the polynomial expression, and then we're going to state that the polynomial is a monomial,
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binomial, or trinomial or none of these. What does that mean? A monomial is a polynomial with
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one term. Mono meaning one. Binomial is a polynomial with two terms,
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and a trinomial is a polynomial with three terms. If there are more terms than three,
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we don't have a special classification for that. We just call it a polynomial.
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This first example is negative two x squared y plus seven x y cubed.
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There are one, two terms in this polynomial meaning that this is a binomial.
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Now let's look at the degree. To determine the degree,
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we look at the degree of each term which is the sum of the powers on the variables.
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The power on the x is two, and the power on the y is one. So this term has a degree of three. Here
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the power on the x is one, and the power on the y is three. So the degree of this term is four. That
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means that the degree of the polynomial is four because that is the largest degree of each term.
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Now let's look at this polynomial. We have three a squared plus two a b plus b squared.
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There are one, two, three terms meaning this is a trinomial. Now let's look at the degree of each term.
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The power on the variable here is two. The power on the a is one, and the power on the b is one, meaning that the degree
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of the second term is one plus one or two. And the power on the b in this third term
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is two. So all three terms have a degree of two, meaning that the degree of the polynomial is two.
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Polynomials that contain like terms can be simplified by combining those like terms.
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Recall that like terms mean that the terms have the same variables raised to the same powers.
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And when you combine like terms, you add or subtract the coefficients to those terms.
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Our first example is three b squared minus eight b squared plus twelve b.
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Let's look for the like terms. This term has a b squared and this term has a b squared,
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the same variables raised to the same power. So they're like terms. This term is not a like term
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to the other two because it has a b raised to the first power. So when I combine these like terms,
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we add or subtract the coefficient. So three minus eight, giving us negative five b
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squared. The variable and its power remain the same. And we still have the plus twelve b. This
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polynomial is simplified. It is also in standard form because the powers are in descending order.
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Let's take a look at our next example. One-half x y plus three x squared y squared plus
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three-halves x y minus x squared y squared. Let's identify the like terms. Here we have
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x and y both raised to the first power, and here we also have a term with x and y raised to the
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first power. So these will be like terms, exact same variables raised to the exact same powers.
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We also have the second term which has an x squared y squared, and the fourth term has
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an x squared y squared. Same variables raised to the same powers means that those two terms
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are like terms. Using the commutative property, I can rearrange these terms
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to make it simpler when I add them or subtract them. So I have one-half x y plus three-halves
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x y plus three x squared y squared minus x squared y squared. Now focusing on our first two
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terms here, which are like terms, we will add the coefficients. One-half plus three-halves
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is four-halves, and four-halves is the same as two. So this gives us two x y
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combining these first two terms. And then combining the last two terms,
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we have a coefficient of three and a coefficient of minus one. So combining three plus a negative
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one gives us two, and the variables stay the same. So we have plus two x squared y squared.
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These two terms cannot be combined because they are not like terms.
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In these next examples, we will add the polynomials. To add polynomials, we simply combine
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like terms. We have the polynomial three x squared plus eight x minus five that we are adding to
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the polynomial two x squared minus six. Let's rearrange this polynomial using the Commutative
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Property of Addition and the Associative Property of Addition, and we'll do some regrouping putting
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our like terms next to each other for clarity. So three x squared and two x squared are like terms,
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so I can rewrite this as three x squared plus two x squared. And then I have a positive eight x. So
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plus eight x. And there is no other like term to eight x, so I will just put eight x.
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And I have a minus five minus six. So I will add negative five and negative six.
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Now adding these like terms here, we keep the variable and the power, and we add the coefficient,
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giving us five x squared. Here we have plus eight x. And negative five minus six is negative eleven. So this
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gives us minus eleven. This is our new polynomial that is the sum of the original two polynomials.
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In our next example, we are adding this trinomial to this trinomial. And again, let's rearrange this
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so that we have the like terms next to each other for clarity. Six a and two a are like terms.
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We have a minus seven b and a plus nine b, so we'll add a negative seven b and nine b together.
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And we have a positive three and a positive five, and so we'll add three and five. Six a plus two a gives us eight a.
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Negative seven b plus nine b gives us plus two b. And three plus five gives us eight.
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To subtract a polynomial, we are going to add its opposite. So let's look at these examples
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where we are subtracting polynomials. Here we have a binomial minus another binomial. So we'll
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rewrite this first as adding the opposite. So we have nine x squared plus four y squared, and
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instead of subtracting this polynomial, we're going to add the opposite of this polynomial
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which is this polynomial times negative one. I'm going to write it this way first.
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So we're adding the opposite of the given polynomial. Well, notice when I have negative one times this polynomial,
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what this does is change the sign of all of the terms in these grouping symbols. And so this becomes nine
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x squared plus four y squared. Minus one times minus two x squared gives me plus two x squared, and minus one
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times ten y squared gives us minus ten y squared. And so now you can see this as adding two polynomials or adding two
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binomials. And combining like terms now, we see that we have nine x squared plus
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two x squared. That will give us eleven x squared. And four y squared minus ten y squared. Four
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minus ten is minus six. So this will give us minus six y squared. Notice that when we add or subtract
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polynomials, we end up with a polynomial. In this case, we ended up with a binomial.
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As we look at this next subtraction problem, we have a polynomial with four terms
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minus a polynomial with three terms. Again, to subtract we just simply add the opposite.
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So we can write this as nine t cubed plus t squared minus twelve t minus four. And now I'm
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going to add the opposite of this polynomial. I'm going to do it a little more efficiently than I
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did here. Remember adding the opposite means that this polynomial is multiplied by negative one
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which changes all of the signs of these terms. So this will become plus a negative t cubed,
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or minus t cubed. Plus a negative of negative t squared, so this will become plus t squared.
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This term will become minus fifteen t. Again take a closer look, all of these
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terms have opposite signs in this addition problem when I rewrote the subtraction
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to addition. And now let's combine like terms. We have a nine t cubed,
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and I have a negative t cubed. Those can be combined to give me eight t cubed. I have a
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t squared and another positive t squared. One t squared plus one t squared gives me two t squared.
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I have a minus twelve t and a minus fifteen t. Minus twelve plus minus fifteen
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gives me minus twenty-seven. So this term will be minus twenty-seven t.
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And I have a minus four still remaining with no other constant terms available.
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And so this is the polynomial answer, or result, of subtracting these two given polynomials.
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In this last example, we're going to simplify this expression where we add and or subtract binomials.
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Five c minus four d is being added to this binomial four c plus eight d, and then we are subtracting this binomial
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two d minus c. Recall that subtracting is adding the opposite. So let's rewrite this problem
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as five c minus four d plus four c plus eight d. And now we're going to rewrite this as an addition of the opposite,
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so plus a negative two d plus c. And this gives us five c minus four d plus four c plus eight d minus two d
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plus c. That's a long expression. Let's find our like terms. I have five c, four
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c, and plus c. Five plus four plus one gives us ten, so this term will be ten c.
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And now we have minus four d plus eight d minus two d. Minus four plus eight
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minus two gives me two d, plus two d. So our final answer is the binomial ten c plus two d.