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Which of the following expressions are degree five polynomials?
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(a) 𝑥 to the fifth power plus five 𝑥 to the sixth power minus two.
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(b) 𝑥 to the fourth power times 𝑦 minus 𝑥 to the fourth power minus three 𝑥 squared.
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(c) 𝑥 cubed 𝑦 squared minus four 𝑥𝑦 squared.
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(d) 𝑥 to the fifth power minus 𝑦 times 𝑥 to the power of negative one.
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And (e) 𝑥 cubed minus two 𝑥𝑦 plus five 𝑥 to the fifth power.
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In this question, we’re asked to determine which of five given expressions are degree five polynomials.
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So we should start by recalling what we mean by a polynomial expression and what it means for a polynomial to have degree five.
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First, we recall a polynomial is the sum of monomials, where a monomial is a product of constants and variables, where our variables must have nonnegative integer exponents.
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We can use this definition to determine which of the five given expressions are polynomials first.
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Let’s start with option (a).
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To determine if this is a polynomial, we first need to check whether each of the five terms are monomials.
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This means they need to be the products of constants and variables and the variables must have nonnegative integer exponents.
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First, we see the only variable is 𝑥 and the exponents of 𝑥 are five and six.
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These are nonnegative integers.
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Next, we need to check that each term is the product of constants and variables.
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Although it’s not necessary, we can rewrite the first term as one times 𝑥 to the fifth power and the third term as negative two 𝑥 to the zeroth power.
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And if we wanted to be even more careful, we could remember that subtracting two 𝑥 to the zeroth power is the same as adding negative two 𝑥 to the zeroth power.
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In either case, we can conclude each term is a monomial.
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So the expression in option (a) is a polynomial.
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We get a very similar story in option (b).
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Each term is the product of constants and variables.
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However, we can see we do have two different variables 𝑥 and 𝑦.
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So we do need to be careful.
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We need to check that the exponents of all of our variables are nonnegative integers.
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In this case, we just write 𝑦 as 𝑦 to the first power.
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So the exponents of our variables are four, one, four, and two.
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These are nonnegative integers.
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The same is true in option (c).
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Each term is the product of constants and variables.
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And we can see all of the variables are raised to nonnegative integer exponents.
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However, the same is not true in option (d).
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We can see we have a term 𝑥 to the power of negative one.
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This is a variable raised to a negative integer exponent.
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So (d) does not represent a polynomial.
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So we can exclude option (d).
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And we can also see that option (e) is a polynomial.
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Now that we’ve concluded expressions (a), (b), (c), and (e) are polynomials, let’s recall how we check the degree of a polynomial.
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We can recall the degree of any polynomial is the greatest sum of the exponents of the variables which appear in any single term.
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And it’s worth pointing out we are only interested in nonzero terms.
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If we have a factor of zero in our term, we can just remove this term and not change the expression.
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We can use this definition to determine the degree of the four polynomial expressions we’re given.
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And to do this, we first note, in the definition of a degree, we take the sum of the exponents of the variables in a single term.
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This means if we’re working with a single variable polynomial, we don’t need to take a sum because there’s only one variable.
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In this case, its degree will just be the largest exponent of that variable which appears in a single nonzero term.
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For example, in expression (a), we can see it’s a single variable polynomial.
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And we can also see that all of the terms are nonzero.
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So we can just check which of the exponents of 𝑥 is the highest.
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And we can see that this is six.
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So expression (a) is a degree six polynomial.
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Therefore, expression (a) is not a degree five polynomial.
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So we can remove this option.
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The remaining expressions (b), (c), and (e) are two-variable polynomials.
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So we are going to need to take the sum of the exponents of the variables in each term.
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Let’s start with expression (b).
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We’ll rewrite 𝑦 as 𝑦 to the first power.
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We need to find the sum of the exponents of the variables in each term.
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In the first term, the exponent of 𝑥 is four and the exponent of 𝑦 is one.
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So we have four plus one is equal to five.
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In the second term, we have a single variable of 𝑥.
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So we just write this as four.
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And in the third term, we only have a single variable of 𝑥.
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So this sum will just give us two.
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The degree of this polynomial is the biggest of these three values, which we can see is five.
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And therefore, the degree of (b) is five.
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So (b) is a degree five polynomial.
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We want to do the same for expression (c).
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We need to add the exponents of the variables in the first term.
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That’s three plus two, which we can evaluate is five.
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We can then do the same for the second term.
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We write 𝑥 as 𝑥 to the first power.
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Then we add the exponents of the variables in this term.
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One plus two is three.
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And then the larger of these two values is the degree of our polynomial.
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We can see this is also five.
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So (c) is a degree five polynomial.
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Finally, we move on to option (e).
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The first term is a single variable of 𝑥.
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So this term has degree three.
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The second term can be rewritten as negative two times 𝑥 to the first power times 𝑦 to the first power.
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And one plus one is equal to two.
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So the second term has degree two.
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Finally, the third term is in a single variable of 𝑥, so its degree is five.
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And since all of these are nonzero terms, the degree is the highest of these values.
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We can see that this is once again five.
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Therefore, option (e) is also a degree five polynomial.
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Therefore, we were able to show, of the five given expressions, only expressions (b), (c), and (e) are degree five polynomials.