Beginning and Intermediate Algebra Chapter 5: …

1 Beginning and Intermediate AlgebraChapter 5: PolynomialsAn open source (CC-BY) textbookbyTylerWallace1 Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative CommonsAttribution Unported License. ( )You are free: to Share: to copy, distribute and transmit the work to Remix: to adapt the workUnder the following conditions: Attribution: You must attribute the work in the manner specified by the author orlicensor (but not in any way that suggests that they endorse you or your use of thework).With the understanding that: Waiver: Any of the above conditions can be waived if you get permission from the copy-right holder. Public Domain: Where the work or any of its elements is in the public domain underapplicable law, that status is in no way affected by the license. Other Rights: In no way are any of the following rights affected by the license: Your fair dealing or fair use rights, or other applicable copyright exceptions andlimitations; The author s moral rights; Rights other persons may have either in the work itself or in how the work is usedsuch as publicity or privacy rights Notice: For any reuse or distribution, you must make clear toothers the license term ofthis work.

2 The best way to do this is with a link to the following web page: is a human readable summary of the full legal code which can be read at the followingURL: 5: - Exponent PropertiesProblems with expoenents can often be simplified using a few basic exponentproperties. Exponents represent repeated multiplication. We will use this fact todiscover the important exponents to multiplication problem(aaa)(aa)Now we have5a sbeing multiplied togethera5 Our SolutionA quicker method to arrive at our answer would have been to just add the expo-nents:a3a2=a3+2=a5 This is known as theproduct rule of exponentsProduct Rule of Exponents:aman=am+nThe product rule of exponents can be used to simplify many problems. We willadd the exponent on like variables. This is shown in the following examplesExample 36 3 Same base,add the exponents2 + 6 + 139 Our SolutionExample 5xy2z3 Multiply2 5,add exponents onx, yandz10x4y7z4 Our SolutionRather than multiplying, we will now try to divide with exponentsExample exponentsaaaaaaaDivide out two of thea saaaConvert to exponentsa3 Our Solution3A quicker method to arrive at the solution would have been to just subtract theexponents,a5a2=a5 2=a3.

3 This is known as the quotient rule of Rule of Exponents:aman=am nThe quotient rule of exponents can similarly be used to simplify exponent prob-lems by subtracting exponents on like variables. This is shown in the base,subtract the exponents78 Our SolutionExample exponents ona, bandc52a2b2cOur SolutionA third property we will look at will have an exponent problemraised to a secondexponent. This is investigated in the following 7.(a2)3 This means we havea2three timesa2 a2 a2 Add exponentsa6 Our solutionA quicker method to arrive at the solution would have been to just multiply theexponents,(a2)3=a2 3=a6. This is known as the power of a power rule of ofaPower Rule of Exponents: (am)n=amnThis property is often combined with two other properties which we will investi-gate 8.(ab)3 This means we have(ab)three times(ab)(ab)(ab)Threea sand threeb scan be written with exponentsa3b3 Our Solution4A quicker method to arrive at the solution would have been to take the exponentof three and put it on each factor in parenthesis,(ab)3=a3b3.

4 This is known asthe power of a product rule or ofaProduct Rule of Exponents: (ab)m=ambmIt is important to careful to only use the power of a product rule with multiplica-tion inside parenthesis. This property does NOT work if there is addition or 9.(a+b)m am+bmThese areNOTequal,beware of this error!Another property that is very similar to the power of a product rule is 10.(ab)3 This means we have the fraction three timse(ab)(ab)(ab)Multiply fractions across top and bottom,using exponentsa3b3 Our SolutionA quicker method to arrive at the solution would have been to put the exponenton every factor in both the numerator and denominator,(ab)3=a3b3. This is knownas the power of a quotient rule of ofaQuotient Rule of Exponents:(ab)m=ambmThe power of a power, product and quotient rules are often used together to sim-plify expressions. This is shown in the following 11.(x3yz2)4 Put the exponent of4on each factor,multiplying powersx12y4z8 Our solution5 Example 12.

5 (a3bc4d5)2 Put the exponent of2on each factor,multiplying powersa6b2c4d10 Our SolutionAs we multiply exponents its important to remember these properties apply toexponents, not bases. An expressions such as53does not mean we multipy 5 by 3,rather we multiply 5 three times,5 5 5 =125. This is shown in the 13.(4x2y5)3 Put the exponent of3on each factor,multiplying powers43x6y15 Evaluate4364x6y15 Our SolutionIn the previous example we did not put the 3 on the 4 and multipyto get 12, thiswould have been incorrect. Never multipy a base by the exponent. These proper-ties pertain to exponents only, not this lesson we have discussed 5 different exponent properties. These rules aresummarized in the following of ExponentsProduct Rule of Exponentsaman=am+nQuotient Rule of Exponentsaman=am nPower ofaPower Rule of Exponents(am)n=amnPower ofaProduct Rule of Exponents(ab)m=ambmPower ofaQuotient Rule of Exponents(ab)m=ambmThese five properties are often mixed up in the same problem.

6 Often there is a bitof flexibility as to which property is used first. However, order of operations stillapplies to a problem. For this reason it is the suggestion of the auther to simplifyinside any parenthesis first, then simplify any exponents (using power rules), andfinally simplify any multiplication or division (using product and quotent rules).This is illustrated in the next few 14.(4x3y 5x4y2)3In parenthesis simplify using product rule,adding exponents(20x7y3)3 With power rules,put three on each factor,multiplying exponents203x21y9 Evaluate 2038000x21y9 Our Solution6 Example (2a4)3 Parenthesis are already simplified,next use power rules7a3(8a12)Using product rule,add exponents and multiply numbers56a15 Our SolutionExample 10a4b32a4b2 Simplify numerator with product rule,adding exponents30a7b42a4b2 Now use the quotient rule to subtract exponents15a3b2 Our SolutionExample (m2n3)3 Use power rule in denominator3m8n12m6n9 Use quotient rule3m2n3 Our solutionExample 18.

7 (3ab2(2a4b2)36a5b7)2 Simplify inside parenthesis first,using power rule in numerator(3ab2(8a12b6)6a5b7)2 Simplify numerator using product rule(24a13b86a5b7)2 Simplify using the quotient rule(4a8b)2 Now that the parenthesis are simplified,use the power rules16a16b2 Our SolutionClearly these problems can quickly become quite involved. Remember to followorder of operations as a guide, simplify inside parenthesisfirst, then power rules,then product and quotient - Exponent )4 44 443)4 225)3m 4mn7)2m4n2 4nm29)(33)411)(44)213)(2u3v2)215)(2a4)41 7)44319)32321)3nm23n23)4x3y33xy425)(x3y4 2x2y3)227)2x(x4y4)429)2x3y23x3y4 4x2y331)((2x)3x3)233)(2y(2x2y4)4)335)(2m n4mn4 2m4n4)337)2xy3 2x2y22xy4 y339)q3r2 (2p2q2r3)22p341)(zy3 zx2y4x3y3z3)443)2x2y2z2 2zx2y2(x2z3)22)4 44 424)3 33 326)3x 4x28)x2y4 xy210)(43)412)(32)314)(xy)316)(2xy)418)3 3320)34322)x2y44xy24)xy34xy26)(u2v2 2u4)328)3vu4 2v2u4v2 2u3v430)2ba2 2b4ba2 3a3b432)2a2b2(ba4)234)yx2 (y4)22y436)n3(n4)22mn38)(2yx2)22x2y4 x240)2x4y3 2zx2y3(xy2z2)442)(2qp3r4 2p3(qrp3)2) - Negative ExponentsThere are a few special exponent properties that deal with exponents that are notpositive.

8 The first is considered in the following example, which is worded out 2different ways:Example the quotient rule to subtract exponentsa0 Our Solution,but now we consider the problemasecond way:a3a3 Rewrite exponents as repeated multiplicationaaaaaaReduce out all thea s11= 1 Our Solution,when we combine the two solutions we get:a0= 1 Our final final result is an imporant property known as the zero power rule of expo-nentsZero Power Rule of Exponents:a0= 1 Any number or expression raised to the zero power will alwaysbe 1. This is illus-trated in the following 20.(3x2)0 Zero power rule1 Our SolutionAnother property we will consider here deals with negative exponents. Again wewill solve the following example two the quotient rule,subtract exponentsa 2 Our Solution,but we will also solve this problem another exponents as repeated multiplicationaaaaaaaaReduce threea sout of top and bottom1aaSimplify to exponents1a2 Our Solution,putting these solutions together gives:a 2=1a2 Our Final SolutionThis example illustrates an important property of exponents.

9 Negative exponentsyeild the reciprocal of the base. Once we take the recipricalthe exponent is nowpositive. Also, it is important to note a negative exponent does not mean theexpression is negative, only that we need the reciprocal of the base. Following arethe rules of negative exponentsRules of Negative Exponets:a m=1m1a m=am(ab) m=bmamNegative exponents can be combined in several different ways. As a general rule ifwe think of our expression as a fraction, negative exponentsin the numeratormust be moved to the denominator, likewise, negative exponents in the denomi-nator need to be moved to the numerator. When the base with exponent moves,the exponent is now positive. This is illustrated in the following 2c2d 1e 4f2 Negative exponents onb, d,andeneed to flip10a3cde42b2f2 Our SolutionAs we simplified our fraction we took special care to move the bases that had anegative exponent, but the expression itself did not becomenegative because ofthose exponents. Also, it is important to remember that exponents only effectwhat they are attached to.

10 The 2 in the denominator of the above example doesnot have an exponent on it, so it does not move with now have the following nine properties of exponents. It isimportant that weare very familiar with all of of Exponentsaman=am+n(ab)m=ambma m=1amaman=am n(ab)m=ambm1a m=am(am)n=amna0= 1(ab) m=bmamSimplifying with negative exponents is much the same as simplifying with positiveexponents. It is the advice of the author to keep the negativeexponents until theend of the problem and then move them around to their correct location (numer-ator or denominator). As we do this it is important to be very careful of rules foradding, subtracting, and multiplying with negatives. Thisis illustrated in the fol-lowing examplesExample 5y 3 3x3y 26x 5y3 Simplify numerator with product rule,adding exponents12x 2y 56x 5y3 Quotient rule to subtract exponets,be careful with negatives!( 2) ( 5) = ( 2) + 5 = 3( 5) 3 = ( 5) + ( 3) = 82x3y 8 Negative exponent needs to move down to denominator2x3y8 Our Solution11 Example 24.