Review of Algebra Review of Algebra Here we review the basic rules and procedures of - PDF document

To add two fractions with the same denominator,we use the Distributive Law:Thus,it is true thatBut remember to avoid the following common error:(For instance,take to see the error.)To add two fractions with different denominators,we use a common denominator:We multiply such fractions as follows:In particular,it is true thatTo divide two fractions,we invert and multiply: b c da bd cad bca ba ba ba bc dac bda bc dadbc bdabc1a bca ba cac ba bc ba bc b1 ba1 bc1 b acac b 12x25x2112x23x92x123x1x32x634x2x32x12x62x212x362x1x56x23x10x56x27x5REVIEW OF ALGEBRA3 We have used the Distributive Law to expand certain algebraic expressions. We some-times need to reverse this process (again using the Distributive Law) by factoring anexpression as a product of simpler ones. The easiest situation occurs when the expres-sion has a common factor as follows:To factor a quadratic of the form we note thatso we need to choose numbers so that and .Factor .The two integers that add to give and multiply to give are and .Factor .Even though the coefÞcient of is not ,we can still look for factors of theform and ,where . Experimentation reveals thatSome special quadratics can be factored by using Equations 1 or 2 (from right toleft) or by using the formula for a difference of squares: 2x27x42x144xs2xr1x22x27x4 x25x24x3883245x25x24crsbr and sxrsx2rsx2bxc  Factoring x y1 1y xxy y xy xxy yx xyxxy yxyx2 xyy2s2t uut 2s2t2u 2us2t2 2x22x6 x2x23 x1x x23x2xx1 x123x6x2x x2x2x3 xx x3 x13 REVIEWOF ALGEBRA The analogous formula for a difference of cubes iswhich you can verify by expanding the right side. For a sum of cubes we have(Equation 2; )(Equation 3; )(Equation 5; )Factoring numerator and denominator,we haveTo factor polynomials of degree 3 or more,we sometimes use the following fact.If is a polynomial and ,then is a factorFactor .Let . If ,where is an integer,thenis a factor of 24. Thus,the possibilities for are and . We nd that ,,. By the Factor Theorem,is a factor. Instead of substituting further,we use long division as follows: Completing the square is a useful technique for graphing parabolas or integratingrational functions. Completing the square means rewriting a quadratic 10 12xx2x2P20P130P112241, 2, 3, 4, 6, 8, 12,bbbPb0Pxx33x210x24x33x210x24PxxbPb0P x216 x22x8x44 x42x4 x2x216 x22x8 ax, b238x222x42x, b5x2252x5x5x, b326x9x32a3b3ab2abb2 3b3ab2abb2 REVIEW OF ALGEBRA5 in the form and can be accomplished by:Factoring the number from the terms involving .cient of .In general,we haveRewrite by completing the square.cient of is . ThusEXAMPLE 10Quadratic FormulaBy completing the square as above we can obtain the following formula for the rootsEXAMPLE 11Solve the equation .With ,,,the quadratic formula gives the solutionsThe quantity that appears in the quadratic formula is called the . There are three possibilities:If ,the equation has two real roots.If ,the roots are equal.If ,the equation has no real root. (The roots are complex.) x332453 25369 10c3b3a55x23x30xbb24ac 2aax2bxc0 2329112x3272x212x112x26x112x26x9911 x2x1x2x1 41 41(x1 2)23 41 4xx2x1axb 2a2cb2 4aax2b a xb 2a2b 2a2 cax2bxcax2b a x cxxaaxp2q6REVIEW OF ALGEBRA These three cases correspond to the fact that the number of times the parabolacrosses the -axis is 2,1,or 0 (see Figure 1). In case (3) the quad-t be factored and is called irreducible.EXAMPLE 12The quadratic is irreducible because its discriminant isnegative:Therefore,it is impossible to factor Recall the binomial expression from Equation 1:If we multiply both sides by and simplify,we get the binomial expansionRepeating this procedure,we getIn general,we have the following formula.If is a positive integer,then 123n aknbnkk12 123 ak3b3abkak1bkk1 12 ak2b2k ab4a44a3b6a2b24ab3b4ab3a33a2b3ab2b3 abab2a22abb2 x2x2b24ac124170x2x2       ax2bxcxyax2bxcREVIEW OF ALGEBRA7 EXAMPLE 13Using the Binomial Theorem with ,,,we haveThe most commonly occurring radicals are square roots. The symbol positive square root of.meansandSince ,the symbol makes sense only when . Here are two rulesfor working with square roots:However,there is no similar rule for the square root of a sum. In fact,you shouldremember to avoid the following common error:(For instance,take and to see the error.)EXAMPLE 14Notice that because indicates the positive square root. In general,if is a positive integer,If is even,then and .Thus because ,but and are not dened. The fol-lowing rules are valid:EXAMPLE 15 3x4 3x3x 3x3 3x x3x a b na nb nab na nb 68 48 23838 2x 0a 0nxnaxna n 1 x2 xx2y x2 y xy 18 2  18 2 9 3b16a9ab a b a b a b ab a b 0a ax2 0x 0x2axa 1 x510x440x380x280x32x25x55x4254 12 x322543 123 x2235x2425k5b2axx25 8REVIEW OF ALGEBRA  To a numerator or denominator that contains an expression such as,we multiply both the numerator and the denominator by the conjugate rad-. Then we can take advantage of the formula for a difference of squares:EXAMPLE 16Rationalize the numerator in the expression .We multiply the numerator and the denominator by the conjugate radicalLet be any positive number and let be a positive integer. Then,by defactorsLet and be positive numbers and let and be anyrational numbers (that is,ratios of integers). Then1.2.3.4.5.In words,these ve laws can be stated as follows:To multiply two powers of the same number,we add the exponents.To divide two powers of the same number,we subtract the exponents.To raise a power to a new power,we multiply the exponents.To raise a product to a power,we raise each factor to the power.To raise a quotient to a power,we raise both numerator and denominator to the power. brar br0abrarbrarsar asarsarasarssrba mnnam (na )m is any integera1nna an1 ana01 anaaana x x(x4 2)1 x4 2x4 2 xx4 2 xx4 2 x4 2x44 x(x4 2)x4 2x4 2 x(a b a b )(a )2(b )2aba b a b REVIEW OF ALGEBRA9 Click here for answers. A REVIEWOF ALGEBRAEXAMPLE 17(c)Alternative solution: x y3y2x 4x3 y3y8x4 4x7y541 3x4 1 x43x43432(4 )323843243 64 8yxx yxyx 2y2 x1y11 x21 y2 1 x1 yy2x2 x2y2 yx y2x2 x2y2 27.28.29Ð48Factor the expression.29.30.31.32.33.34.35.36.37.38.39.40.41.42.43.44.45.46.47.48.Simplify the expression.49.50.51.52. x31 x29x35x26x x2x12x21 x29x82x23x2 x24x2x2 x23x2x33x24x12x35x22x24x32x223x60x33x2x3x34x25x2x32x2xx3274t212t94t29s2t31x210x256x25x68x210x39x2362x27x4x22x8x2x6x27x65ab8abc2x12x311 11 1x11 c1 11 1Ð16Expand and simplify.1.2.3.4.5.6.9.10.11.12.13.14.15.16.17Ð28Perform the indicated operations and simplify.17.18.19.20.21.22.23.24.25.26. bc b ac2r s2 6tx yxy 2 a23 4 b2u1u u11 x11 x11 x52 x39b6 3b28x 21xx2212x23x1t522t3t1y46yy23x22x12xx124x1x753t4t222tt34x2x25x22x184x243a43xx2xx52x2y6abac 85.86.87.88.89.90.91.92.93.94.95.96.97.98.99.100.101Ð108Rationalize the expression.101.102.103.104.105.106.107.108.109Ð116values of the variable.109.110.111.112.113.114.115.116. 4x1 22 xx xy1 1y1 x1y1xy16a 161a 16x24 x2x2 xx2x x2x x23x4 x1 x y 2 35 2h 2h hxx 8 x4(1x )1 x1x 3 x94r2n1 4r1 4t12 23 x5 4x3 81 (t )5(4a )35y6 x5y310352x2y4326443125239615312x1y1 xy1a3b4 a5b5ana2n1 an2x92x4 55.56.57.58.59.60.Solve the equation.61.62.63.64.65.66.67.68.Which of the quadratics are irreducible?69.70.71.72.Use the Binomial Theorem to expand the expression.73.74.75.76.77Ð8277.78.79.81.82.83Ð100Use the Laws of Exponents to rewrite and simplifythe expression.83.84. 53a 16a4b3  x3y 432x4 42 32 354 32 2 3x25x214ab7ab6x23x63x2x62x29x42x23x4x33x2x10x32x102x27x203x25x10x22x70x29x10x22x80x29x1003x224x504x24x2x23x1x25x10x216x80x22x5x x2x22 x25x4 63.64.65.66.67.68.Not irreducible (two real roots)77.78.79.80.81.82.83.84.85.86.87.88.89.90.91.92.93.94.95.96.97.98.99.100.101.102.103.104.105.106.107.108.109.False110.False111.True112.False113.False114.False115.False116.True x2x x2x 3x4 x23x4 xx y xy35 22 2h 2h x24x16 xx 81 x x1 x 3rn2t14 s1241 x18t52a34y65x3 y95622 x3y61 25625253 1 3 xy2 2 ba2n316x102603262a4a2bb x2y2x1 38243405x2270x490x615x8x10x84x66x44x21 21a2b57ab6b7a77a6b21a5b235a4b335a3b4a66a5b15a4b220a3b315a2b46ab5b61, 12 1, 15 2733 4513 6122 985 1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23.24.25.26.27.28.29.30.31.32.33.34.35.36.37.38.39.40.41.42.43.44.45.46.47.48.49.50.51.52.53.54.55.56.57.58.59.60.61.62. 2)25 4(x5 2)215 4x8216x124x26x4 x124x2 x29xx2 x4x1 x82x1 x2x2 x2x232x234x354x113x12x2xx12x323x92t322t3st3st12t1x523x2x34x3x19x222x14x42x32x6158c2x16x232x 2xc c2a2 b2 tx yx y2b234a2 a2b2u23u1 u12x x213x7 x22x2b14xx42x3x22x12x35x2x12304y5y69x244x24x1x3x22x12x225x73t221t22x26x34x86a4x3x22x210x2x3y53a2bc Here we review the basic rules and procedures of algebra that you need to know inThe real numbers have the following properties:(Commutative Law)(Associative Law)(Distributive law)In particular,putting in the Distributive Law,we get(c)If we use the Distributive Law three times,we getThis says that we multiply two factors by multiplying each term in one factor by eachterm in the other factor and adding the products. Schematically,we haveIn the case where and ,we haveSimilarly,we obtain ab2a22abb2 ab2a2baabb2dbcaabdabdabcabdacbcadbd 43x243x6103x2t7x211144t2x22t REVIEWOF ALGEBRA Review of Algebra