Definitions, Concepts, and Formulas | Examples | ||||||||||||||||||||||||||
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P.1 The Real Numbers and Their Properties
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7, 18, 53 0, 3, 8 −5, −1, 0, 17, 23 −157, −63, 0, 12, 0.25, 0.1¯23 −√6, −√2, √3, π, 5.020020002… −11, −3.57, 0, √2, 574 3<7, −√2<√2,0<2 |3|=3; |−4|=−(−4)=4 3+√7 3+5=5+3; −5⋅7=7⋅(−5) [5+(−3)]+7=5+[(−3)+7] (5⋅6)⋅15=5⋅(6⋅15) 2(3+7)=2⋅3+2⋅7 −5+0=0+5=5; 7⋅1=1⋅7=7 15+(−5)=0; 15⋅115=1 0⋅3=0=3⋅0 |
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P.2 Integer Exponents and Scientific Notation
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a4=a⋅a⋅a⋅a 50=1, π0=1, (−17)0=1 5−3=153 73⋅72=73+2=755652=56−2=54(a3)4=a3⋅4=a12(3u)4=34u4=81u4(x2)3=x323=x38(3y)−2=(y3)2=y232=y29 |
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P.3 PolynomialsAn algebraic expression of the form anxn+an−1xn−1+⋯+a2x2+a1x+a0,
where n is a nonnegative integer, is a polynomial in x. If an≠0, To add or subtract two polynomials, we add or subtract the coefficients of the like terms. To multiply two polynomials, distribute each term of the first polynomial to each term of the second polynomial and then combine terms. Special Products (x+a)(x+b)=x2+(a+b)x+ab(A+B)(A−B)=A2−B2(A+B)2=A2+2AB+B2(A−B)2=A2−2AB+B2 |
5x3−2x+7 (5x3−2x2+3x+1)+(2x3+5x−7)=5x3−2x2+3x+1+2x3+0x2+5x−7=(5+2)x3+(−2+0)x2+(3+5)x+(1−7)=7x3−2x2+8x−6(3x2+5x−3)−(2x2−x−7)=3x2+5x−3−2x2+x+7=(3−2)x2+(5+1)x−3+7=x2+6x+4
(x+2)(x−3)=x2−x−6(x+2y)(x−2y)=x2−(2y)2=x2−4y2(2x+3y)2=(2x)2+2(2x)(3y)+(3y)2=4x2+12xy+9y2(3x−4)2=(3x)2−2(3x)(4)+42=9x2−24x+16 |
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P.4 Factoring PolynomialsFactoring formulas:
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4x2−25=(2x+5)(2x−5)4x2+12x+9=(2x+3)29x2−12xy+4y2=(3x−2y)2x3−8=(x−2)(x2+2x+4)8x3+27y3=(2x+3y)(4x2−6xy+9y2) |
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P.5 Rational ExpressionsThe quotient of two polynomials is called a rational expression.
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23x⋅5y=103xy
3x4y÷5x26y3=3x4y⋅6y35x2=18xy320x2y=9y210x
2x+3y=2y+3xxy; x3−y2=2x−3y6 |
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P.6 Rational Exponents and Radicals
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4√81
823=(3√8)2=22=4
√32=3; √(−3)2=3 |