Chapter 3 Section 6 Lesson Polynomials


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1 Chapter Section 6 Lesson Polynomials Introduction This lesson introduces polynomials and like terms. As we learned earlier, a monomial is a constant, a variable, or the product of constants and variables. Some eamples of monomials follow: 6 9, 7, y,, 14 y z A polynomial is a monomial or the sum or difference of any number of monomials. The monomials listed above are eamples of polynomials, as are the following: y + y z z Some eamples of epressions that are not polynomials follow. Epression Why isn t it a polynomial? A polynomial cannot contain a variable in a denominator. A polynomial cannot contain a variable under a radical sign. 6 + There is a variable in a denominator. Question: Is 5 7t t +1 a polynomial? 5 Answer: Yes. It is the sum of the monomials 7t, t, and 1. Question: Is 4 9 t + st + a polynomial? Answer: Yes. It is the sum of the monomials t, 4, st, and 9. Page 1 of 5
2 Question: Is y a polynomial? 4y + 1 Answer: No. If a fraction has a variable in its denominator, it is not a monomial. So, not a monomial and the given epression is not a polynomial. 4y + 1 is When a polynomial is written in its usual way, as a sum of monomials, then each monomial is a term of the polynomial. Monomials are terms, but so are many algebraic epressions that are not monomials. In general, a term in an algebraic epression is a monomial or any epression that is a product of factors. If two or more terms in a polynomial have the same variables raised to the same power they are called like terms. Eample: In the polynomial 5 y 9y + 7 y 4, the first and third terms, 5 y and 7 y, are like terms. Notes: Like terms can be added. This is called combining like terms. When you combine like terms, you add the coefficients of each term and leave the variable parts unchanged. Thus, 5 y 9y + 7 y 4 = 1 y 9y 4. The distributive property, c c ( ) a + b = a + b c, justifies combining like terms. Consider the like terms that were combined above. Using the distributive property: 5 y + 7 y = 5+ 7 y = 1 y. ( ) Polynomials with two terms are called binomials. Polynomials with three terms are called trinomials. Eample A Simplify 8 y y + y 7 by combining like terms. 8 y y + y 7 = 11 y y 7 Page of 5
3 Eample B Combine the like terms in the polynomials below. a) 9r + r + 4r 6 b) 4t + t 6t t. a) b) 9r + r + 4r 6 = 9r + r + 4r 6 = r 5r 4t + t 6t t = t In this eample, the t terms add to. Question: Combine like terms in the polynomial r + s + 5r 7s. Answer: Identify the like terms: r + s + 5r 7s Then combine like terms: = 8r 5s We saw earlier that the degree of a monomial with only one variable is simply the degree of that variable. When a monomial has more than one variable, the degree of the monomial is the sum of the eponents of all the variables of the monomial. Eamples: The monomial y has a degree of 5, since + = t has a degree of st = s t has a degree of. 1 6y = 6 y has a degree of. Note: Recall that constants such as 1, 5, and 1. 9 are considered monomials. Such constant, but nonzero, monomials have degree zero. This makes sense in terms of the eponent rule, = 1. For eample, using this peculiar way of writing the number one, we can show that the degree of the constant monomial known as the number is indeed zero: = 1 = =. The degree of is zero, as is the degree of any constant. The only monomial whose degree is undefined is the monomial. The degree of a polynomial in standard form is the highest degree of any of its terms (assuming its like terms have all been combined). Page of 5
4 Eample C What is the degree of each monomial in the polynomial 4 y + y 4? What is the degree of the polynomial? a) The degree of the first term, 4 = 4, is 1. 1 b) The degree of the second term, y = y, is. c) The degree of the third term, y = y, is also. 1 1 d) The degree of the fourth term, 4 = 4 1= 4, is. e) Since is the highest degree of any term in the polynomial, the degree of this polynomial is. This image represents an animation that can only be seen in the course online. Eample D What is the degree of y y + y 5+ 7y? The degree of this epression seems to be 4 because the highest degree of the terms shown is 4. But wait The like terms have not been combined! When y and y are combined, they cancel and the result is zero. So, the epression reduces to the polynomial y y. Now, the highest degree of these terms is. The polynomial actually has a degree of. A Word to the Wise Always combine like terms before deciding the degree of a polynomial. Etended Eample 1a Find the degree of the polynomial yz yz 9y z + yz + 9y z +. Hint: Combine like terms. Step 1: yz yz 9y z + yz + 9y z + = 5yz yz + Hint: Find the degree of each term. Step : 1 1 5yz = 5y z has degree 1+ 1 = ; yz = y z has degree = ; = has degree =. Hint: Select the greatest of those degrees. Answer: The polynomial has degree. Note: Once the like terms were combined, the polynomial above turned out to be a trinomial (a polynomial with three terms). rd degree Page 4 of 5
5 Etended Eample 1b Find the degree of the polynomial yz 9 4 y z + 5 yz yz + y z 5 yz 9 y z. Hint: Don t panic! Many of the terms cancel once you combine like terms. Step 1: yz 9 y z yz 9 + yz + y z 4 5 yz 9 y z = yz + yz Hint: Find the degree of each term. Step : 1 yz = y z has degree + 1+ = 5; 1 1 yz = y z has degree = 4; Hint: Select the greatest of those degrees. Answer: The polynomial has degree 5. th Note: Once the like terms were combined, the polynomial above turned out to be a 5 degree binomial (a polynomial with two terms). Eample E Determine the degree of each polynomial below. 4 a) b) 4 9 n + mn + c) 8 s t + d) 6 yz + 4yz + 6y z e) 9n 6 a) 4 b) c) This is not a polynomial, since 8 s is not a monomial. t d), since the second and fourth terms cancel out. e) 1, because the second term equals zero and has no degree. We end this lesson by classifying the polynomials given as eamples on page 1, noting the degree of each polynomial and whether it's a monomial, binomial, or trinomial: degree monomial degree 1 monomial degree 1 binomial degree trinomial 9 5 y + y z z.. degree 1 trinomial End of Lesson Page 5 of 5
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