# Getting more involved 93 writing what is the

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Chapter 2 / Exercise 43
Intermediate Algebra: Connecting Concepts through Applications
Clark
Expert Verified
Getting More Involved93.WritingWhat is the difference between the equations(x5)2x210x25 and (x5)2x225?94.WritingIs it possible to square a sum or a difference withoutusing the rules presented in this section? Why should youlearn the rules given in this section?In This SectionU1VDividing Monomials U2VDividing a Polynomial by aMonomialU3VDividing a Polynomial by aBinomial4.8Division of PolynomialsYou multiplied polynomials in Section 4.5. In this section, you will learn to dividepolynomials.U1VDividing MonomialsWe actually divided some monomials in Section 4.1 using the quotient rule for expo-nents. We use the quotient rule here also. In Section 4.2, we divided expressions withpositive and negative exponents. Since monomials and polynomials have nonnegativeexponents only, we will not be using negative exponents here. E X A M P L E 1Dividing monomialsFind each quotient. All variables represent nonzero real numbers.a)(12x5)(3x2)b)24xx33c)120aa22bb24Solutiona)12x5(3x2) 132xx254x524x3The quotient is 4x3. Use the definition of division to check that 4x33x212x5.b)24xx332x332x02 1 2The quotient is 2. Use the definition of division to check that 2 2x34x3.c)120aa23bb245a32b425ab2The quotient is 5ab2. Check that 5ab2(2a2b2)10a3b4.Now do Exercises 1–18If abc, then ais called the dividend,bis called the divisor, and cis called thequotient.We use these terms with division of real numbers or division of polynomials. U2VDividing a Polynomial by a MonomialWe divided some simple polynomials by monomials in Chapter 1 using the distributiveproperty. Now that we have the rules of exponents, we can use them to divide polyno-mials of higher degrees by monomials. Because of the distributive property, each termof the polynomial in the numerator is divided by the monomial from the denominator.
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Chapter 2 / Exercise 43
Intermediate Algebra: Connecting Concepts through Applications
Clark
Expert Verified
306Chapter 4Exponents and Polynomials4-52E X A M P L E 2Dividing a polynomial by a monomialFind the quotient.a)(5x 10) 5b)(8x612x44x2)(4x2)Solutiona)By the distributive property, each term of 5x10 is divided by 5:5x51055x150x 2The quotient is x2. Check by multiplying: 5(x2) 5x10.b)By the distributive property, each term of 8x612x44x2is divided by 4x2:8x641x22x44x248xx26142xx2444xx222x43x21The quotient is 2x43x21. We can check by multiplying.4x2(2x43x21)8x612x44x2Now do Exercises 19–26Because division by zero is undefined, we will always assume that the divisor isnonzero in any quotient involving variables. For example, the division in Example 3is valid only if 4x20, or x0.U3VDividing a Polynomial by a BinomialDivision of whole numbers is often done with a procedure called long division.Forexample, 253 is divided by 7 as follows:36QuotientDivisor7 253Dividend2143421RemainderNote that the remainder must be smaller than the divisor anddividend (quotient)(divisor)