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(3x ²³ + 4x ²-5) 2x²+x-1 3x²+4x² -5 3x³ + 6x²+x-6 Arrange the forms of each polynomia in descending order with like terms in the same column. A Combine the terms in each column. Example 2-Add. Use a horizontal format Solution: (3x³-7x+2)+(7x²+2x-7) 4 (3x³-7x + 2) + (7x² + 2x-7) = 3x² + 7x²+(-7x + 2x)+(2-7) 3x²³+7x²-5x-5 Use the Commutative and Associative Properties of Addition to rearrange and group like forms *Combine like terms, and write the polynomial in descending order. Ⓒ The opposite of the polynominalx ² - 2x +3 can be written: The opposite of a polynomial is the polynomial with the sign of every term changed. Ex:-(x²-2x+3)=-x²+2x-3 Polynomials can be subtracted using either a vertical orghorizontal format. To subtract, add the opposite of the second polynomial to the first Example 3-Subtract. Use a vertical format. (-3x+7) - (& ²+3x-4) Solution: The opposite of -8x²+3x-4 is 8x²-3x+4₂ 4-3x²-7 5x²-3x-3 Write the terms of each, polynomial in descending order with like terms in the O Combine the terms in cach column. ( Example 4-Subtract. Use a horizontal format. (Sx²-3x+4)-(-3x²³-2x+8) Solution: (5x²-3x+4)-(-3x³ + 2x-8) = (5x²-3x+4) + (3x³ + 2x-8) = 3x² + 5x ² + (-3x + 2x) + (4-8 -3x + 2x) = 3x²³+ 5x²-x-4 +3X²-4x -2x²-3x #13-33 (13) (x² + 7x) + (~3x² - 4x) 19, (3y²-2y) + (Sy² +6y) , (y² +4y) + (-4y-8) x² + 7x 4y² +44 (3x²+2x) +(6x-24) 43x² +9x 4x²-7x+4 2x² +X-10 3y²-2y Rewrite subtraction as addition, of the appasite + Sy² + 6y 8√²+4y Dearrange and group like terms *Combinelike terms. Write the rolynomial indescending order. + 6x-24 +3x²+x+8 3x²+15x-24 5x²+7x+ 20 19 (x²-7x+4)+(2x²+x-10) 20, (3y²³+ y² + 1) + (-4y²-6y-3) 21 (2a²³ - 7a+ 1) + ², 3y ³²+ y² + 1₂ (-3a²-4a+1) (7) (2x² + 6x +12) + (3x²+x+8) 18), (x²+x+5) + (3x²-10x +4) 4x²+x+5 42x²+6x +12 +3x²-10x +4 x²+2x²-6x-6 -Y²³+ y ² - 6y=2] 24 (5r²-6r² + 3r) + (r²-2r-3) Ⓡ3 (4x²+2x) + (x²+6x) 5r³-6r²+3r 4x²+2x+ 42a²³-70+ 1. +(-30²-4a+1) [20²³-30²-||a + 2 + X² + 6x 1²-25-3 Sr³-5p²+r-3 Sx + 8x + 24 (-3y² + y ) + (4y²+ 6y) (23) (4x ² - 5xy ) + (2x² + (xy-4y²) (26) (2x²-4x² -3y² + y +4√²+64 • 4x²-5xy +3x²+6xy-4x² 17x² + xy = 4y ²] (6x²-2xy + 4y²) ·2x²-4x² +6x² - 2xy + ²4 18x²-2xx Ly² + 7y (217) (2a²-7a+10) + (a1²=+ 4a-7) (28) ₁ (-6x² + 7x+3) + (²x²+x+3) 4 5-6x² + 7x+3. 2a²-7a+10 + a² +40+7 30²-39+17 (29) 12 (5x² + 7x-7) + (10x² - 8x+3) @₁ (3√² + 4y + 9) + (2x ² + 4y -21) 5x³+7x-7 3y²³ + 4y +9 +2√² +44-21 +10x²8x+3 [5x²³ + 10x²-x-4] 3₁ (2₁²-5r+7)+(3r²³-6r) 2r²-5r+7 +3r³ -6r 353³+ 2²-116+7 +3x²+x+3 -3x²+8x+6 (33) (3x² + 7x+10) + (-2x³ + 3x+1) +(-2³ 3x² + 7x + 10 + 3x + 1) -2x ³ + 3x² + 10x + 11 3y ³ + 2y ² + 8y-12 1 (32) (34 ³² + 4y + 14) + (-4₁² +21) 4 3y ³+ 4y +14. +(-4/²² ·10x 3√³-4√² + 4y +35 #39-57 (038) (39), (x²-6x)-(x²-10x) ) - (x²-10x) ¹1 (2√²-4y)-(-y² + 2) (13) (x² - 2x + 1)-(x²+5+8) x-2x-1 x²-6x 2y²-4y √²- 2 [x² = 4y=21 -X +5+8 13x-7 (15) (4x²+ 5x + 2)-(-3x²+2x+1) 7 (2y³ + 6y-2)-(y²+ y² + 4) + (y² -100 xy) - 2₂ $√² +61-2 -(-3,² 4₂4x²³² + 5x + 2. + 2x + 1) L (2y² + 3xy) y²-100xy 2 √²+ 3xy E103²-² 12 (31₁ (3x²+x-3) -(x²+x-2) 63) (-2x³+x-1)-(-x²+4x-2) 43x²+x-3 x²+4x-2 2x²+3x-1 4-2x³+X-1 E2x²+x²-3x+1 (53), (49²-29+1)-(9²³-2a+3) 57, (4y ³-y-1)-(2y²-3y+3) 44y ³-y-1 2²-3y +3 14√³-2y² + 2√-4 44a³-20+1 -a²³² +26-3 139³-2 (#9-37 $43-93 odds) Section 7.2: Multiplication of Monomials 1. Recall that in the exponential expression xx is the base and 5 is the exponent 2. The exponent indicates the number of times the base occurs as a factor. 3. The product of exponential expressions with the same base can be simplified by writing each expression in factored form and writing the result with an exponent which is shown below: 3 factors 2 factors x^²- X* = (x-*-* )-(x -x) 5 factors X.X.X.X.X EX 4. Adding the exponents results in the same product as shown below: 2 3+2 =X =x² RULE FOR MULTIPLYING EXPONENTIAL EXPRESSIONS If m and n are integers, thenx.xn-xmth, EXAMPLES: In each example below, we are multiplying two exponential expressions with the same base. Simplify the expression by adding the exponents. 7 4+7 ⒸX-X-X-X11 @Y₁ Y ² = y ¹+²=y6 3a²·a²·a=a²+6+²2 9 Example 1-Hultiply. Solution: (3x¹y) (2-3)(x-x²) = (2) (***) (4-4) Properties of Multiplication to rearrange 1+1 = 6x³y² and group factors. Multiply variades with the same • base by adding the exponent... t 5. A power of a monomial can be simplified by rewriting the expression in factored form and then using the rule for Multiplying Exponential Expressions as shown below. (x²)³=x²-x²-x² =X² EX 212+2 |(x²y³)² = (x *y ²) ( XY a (3 (²42) 4+4 6 EXY 6. Please note that multiplying each exponent inside the parentheses by the exponent outside the parentheses gives the same resultas 1. shown below: (X²) ³ = x ² + ² = X ²) -1. (x ² √ ² ) ² = x 4.2 √ ³·2=√x 8y 6 RULE FOR SIMPLIFYING POWERS OFEXPONENTIAL EXPRESSIONS If m and n are integers, then (x)=xmn, EXAMPLES: 1.3 Each example below is a power of a product of an exponential expression. Simplify the expression by multiplying the exponents. 1) (x²³²) ² = X² 2(13) 434- =X10 12 RULE FOR SIMPLIFYTNG POWERS OF PRODUCTS If m.n. and pare inkgers, then (xmyn)²=X²yn² EXAMPLES Each example below is a power of a product of exponential expressions. Simplify the expression by multiplying each exponent inside the parentheses by the exponent outside the parentheses. 6 (5³) 65²68³6d¹9. (20 ²6³) = 31.3 3 a²b² = 27a²b²³ Example 3-Simplify. @9) Solution: (-2x) (-32) (29) 3.6 -Y₂ :(-2x) (-3) ³x² (-2x) (-27) x ³/4 6 = [-2 (-27)] (x+x²) y6 = 4..6 AMultiply each exponent in-3xy ² by the exponent outside the parentheses. Simplify (3) *Use the Properties of Multiplication Ho rearrange and group factors. to Hultiply variable expressions with the some base by adding the exparents. #9-37 (X)(2x) = 2 · X-X = 2 • X ²²₂² = 12x²1 (-3√) (1) == 3 ⋅ Y⋅Y = -3-y ¹²¹² = - 3y² (3x)(4x)-12x² 3+2 49,5 49 (13 (-2a²³) (-3a¹)= (-2--3). (a·a·a)-(a·a·a·a) = -6a³¹4 = -60° (506) (-2)=(5-2). (a·a·a·a·a·a)· (a·a·a·a·a) = -10a5+5=-100 11 (^²y^²) (x²³x *) = (x-x) • (X.X.X.X.X) · (Y· Y · Y · Y) · (Y· Y · Y ·Y) = X ² * -2x¹)(Sx²)=(-2-5). (x-x-x·x) (X-X.X.XX) · (y) = -(Ox (-30²) (20²6¹)= (-3.2)· (9·a·a)· (a.a) · (b·b·b·b)=-6₂³426" - 6a³b4 ²b²ab²) = (a·a) · (a) · (b·b·b·b) · (b·b·b)=q²+¹b²4+ [³67 ) (xY²) = (x ·x) • ( y. y. y⋅y). (xy ·xy.xy-xy-xy-xy-xy) = X = (2·--3)⋅ (xY) · (X-X)⋅ (Y⋅Y-Y⋅Y) = -6X²² (22) (-3a²b) (-2ab²³)= (-3--2-a.a) (b) (ab-ab-ab)=6a²+¹b² (23) (x²-yz) (x²y¹) = (x+x) · (x:x)-(y) (y-Y-Y-Y)-(YZ)=x² ¹² y ³¹ ¹Z-X³¹Y/Z (24 (-ab³²c) (a²6³) = -a²²b²c-ab²c]. 2+1, 4+7 (2xy) 1+4 2+1 3+1 3/4 2+21+4 (a¹²b³) (ab²c¹) = (a·a)·(a) (bb)· (b·b·b) · (c.c.c⋅c) = 0 2² +¹ 6 2 +³3 4 2+2 4+1 z 5 3.11 b³cy a²³bc² 2+32+6 ) (a²³b²)= (-a.a)· (a·a·a) · (b·b) · (b·b·b·b·b·b) = _a² 473 ху ху-ху-ху 7 XY₁-XY⋅ XY ) = x^²y^² 2+1/3+2 b) = (-6)· (a·a·a) (a.)· (b) = -68²b-1-6a² (-4ab²) = (2-4)· (a.a) (a), (b·b·b). (b.b)= -82²¹¹6 3+2√ [80 -5y²/2) (-8,² = ³) = (-5²8) · (Y⋅Y⋅Y⋅y)· (y⋅Y-Y⋅Y⋅Y⋅Y) · (Z•Z•Z•Z•Z-z)-(z) 2 21¹s 140x²0=² 8 E 32 (3x²y) (-4x²)=(3--4)-(x-X)⋅ (x) · (Y) (Y-Y) = ~12x²+2√²+2, 23 (10ab²) (-2ab) = (10--2) (ab) (ab⋅ab) ==20gb¹+² 200b² 12+13_1+1 (³²1) (12) (xyz)=(xx); (1) -(1)-(x) -(1)-(. ). (z)=x²+²x+³² ²¹² (xy^²Z) (x²y) (z ²y ²) = (x) · (Y⋅y)<(Z). (x-x) - (y) · (2-2) - (y⋅Y)=X (3²) (35) #43-93 (Odd) + x ³y = Z²³ 66 (2x²y³) (3xy) (-5x²³ y ² ) = (-2·3₁-5)⋅ (x·x); (x) (*.xx); (x-y-x). (Y). 2+1+3 v3.3-v9 172 30x 37 (4a²b)(-3a²b²) (a³b²³) = (4.-3) (a·a)·(aaa) · (a·a·a·a·a) · (b) · (b·b·b·b) 4·(b-b) = -12a²+3+5 b4+2=-=-=-120 1⁰b7 10 (43) X³³-X² (45) (x²) 14 X (2²)3-22-3 4³-64 2-2 (-2²) ³ = (4) ³²-164 (-x²) ² (2x)² = (2x²) (2x²)³=-=-8x 2.3 -=-{X² (3x²y)² = (3x 2-2) =19x" Exa -8x6 6 2 YZ 1-2 व4 2 2+3-2+1 'Z (9²) (30²)² =a².3³.a²²3-3¹). (9²) (as)-(3)(a) (a²+6)=√27a8) 63 (-2x) (²x²³) = (2-2).(x). EX- •Y ²³ = x ² yx³y²³ 2+6 3+1 (ab²)² (ab²)=(a²). (b²²) ab²-a².b₁.ab²=a²+¹b4+2=5736 3-3 3 (-2x) (-3xy²)² = (2x) (-3)^(x¹). (y²^²) = (-2) (2) (x) (x²) (y) = -181² = y²4 4-18x²³y" (73) (ab ²) (-2a²b)³ = (a) (b²)· (-2) ³ (a²³^²)+(6³)= (-2)³• a·b ². aº. b²³ -2³ab²aa³=-8ab²a²b²³-80¹161 4+6₂ ²+3 = -80,76² •(6³) = (2)· (3.3²). (9³) · (a6). (6³) (-6). (a)· (a²³)· (as)· (b²³) = -54 a ²³ +66 ²³ 54a9b³ 3+61 4.(-2)· (3)· (9)·( 19 (-3ab)²(-2ab)³ S 85 (-3)². (@²) · (6²) · (~20b) ³ (9)· (a²). (6²)· (206)³ [ 9a²b²-(-2) ³. (a³). (b³) (9)·(-2)²³-a²-b²a³-b³ (-4)-(2)-(2²)-(9²)-(6²)-(9²)-(6²) -9.2.4-9².6².0².6³ -9-8-a²-b²-a²b -9a²+3b2+3 (8)=-72a²b²/ (6x) (2x²) + (4x²) (Sx) @ (3a²b³²) (lab) - (9ab²2) (a²b) (6x) (5x) + (2x²) (4x²) (3-2)· (9²+1). (b²+1)-(9ab²). (a²b) (3-2) (a) b³-9ab²a²b 30x² +8x 6a³b²-9ab²a²b 46a³b²³-9a¹+2√²+1 6a³b³-9a³b³-3a³b²³ 191g₁ • a^=b² n n √²2) - (2 x ²) (1) (x²)(y^²) @ (a²) ^ -a²0 ) ( ) 2n (89) y²³) (3x ²y ²) - (2x ³ x ) <(x ² x ²) 15x5y5-2x3+2 4+1 15x³y ¹5- 2x³5³ 13x55 Ⓒ40² (206)²-5b²(a²b) (4a²)-(23)-(a³)-(6³)-(56²)-(as) (b) (4-2-2-a²a³-b³-5b²a³b 4-2-4-a²-a³-b³-Sb²a³b 4.8-a²-a³b²-5b²a³b 4-8a²+3b²³-Sb²a³b. 32ab³-5b²a³b -2xy - x ³²2 y ²³-3x³ -2xy-x²y³²-3x²xy ²² -2x²x²y³ -2x ²²+6y²+³ - 3x²³x²y -2x³y² - 3x²x²y" (Xy3)₂ -3x³y² - 2x² 2n Section 7.3 Multiplication of Polynomials 1. To multiply a polynominal by a manormal, use the Distributive Property and the Rule for Multiplung Exponenbal Expressions Example 1-Hultiply. A.-2x (x²-4x-3) AUse the Distributive Property. -2x (x²)-(-2x) (4x)-(-2x) (3) Use the Rule for Multiplying Exponential Expressions AUse the Distributive Property. I Use the Rule for Multiplying Exponential Expressions AUse the Distributive Property and the Pule for Mulhphying Exponential Expressions -2x³+x²+6x B. (5x+4) (2x) 5x (2x) + 4 (-2x) 110x²-8x C. x²(2x²-3x+2) (x²³) (2x²) - (x²) (3x) + (x²) (2) Dy-3x+2x² 2. Multiplication of two polynominals requires the repeated application of the Distributive Property 7 "(y-2) (y² + 3y + 1) = (y-2) (y²) + (y-2) (3y) + (y-2) (1) y²-2y² + 3y²-6y+y=2 _y²³ +y²²-54-21 3. A convenient method of multiplying twopolynomials is to use a vertical format similar to that used for mulpplication for whole numbers. →y² + 3y+1 y-2 1-2y²-6y-2 y²³²+ 3y² + y [√³+√²-54-2] Example 2-Multiply. Solution: 26³-b+1 Mulhply each term in the trinomial by-2. He Mulholy each form in the tamainal by y. I Like kaims must be in the same column. Add the terms in each column. 42.4 2b+3 6b²-36+3 -26² +26 St Halldy 26²-bi1 by 2 Multiply, 2b³-b+1 by 2b. Arrange the Terme Todescerning order And the arms in eachic duron. 4b²+6b²-2b²-b+3 4. It is often necessary to find the product of two binomials. The product can be found using a method called FOIL (First, Outer, Inner, Last). Example-multiplying two binomials using the FOIL method Multiply: (2x+3)(x+5) L₂ (2x+3)(x+s) - 2x+x=2x² (2x+3)(x+5)-2x-5=10x (2x+3)(x+5)-3-x= 3x (2x+3)(x+5)-3-5=15 (2x+3)(x+5)= F O IL 2x² +10x+3x+15 2x+3x+15 Example S-Multiply Solution: (3x-2y) (x+4y) Example 4-Multiply. Solution: (4-3) (3x-2) F 4x (3x) + 4x (-2) + (-3) (3x) + (-3) (-2) 12x28x9x+6 12x²-17x+6 Alfolliply the first tams. Helluliply the Quer terras. Morlify the Immer forms Paliph the Latterns. Add the products. & Combine the Terms 3x² + 12xy - 2xy - 8y² F I 3x (x) + 3x (44) + (-2y) (x) + (-24) (4) Method. = =q²-6² a and the second lemm in each binamal is b. 7. The expression (a+b) is the square of a binomial. THE SUM AND DIFFERENCE OF TWOTERMS (a+b) (a+b)=a²-abtab-b² Square of 1 first term Use the FOIL Square of (second term Hellod *Combine like erms = = 3x² +1 + 10xy-8y ² 5. The expression (atb) (a-b) is the product of the sum and the difference. of two terms. Use the FOIL 6. The first binomial in the expression is a sum and the second binomial is a difference. The two terms are a ond b. The first term in each binomial is Combine like Example 6-Multiply. Solution: (2x+3)(2x-3) THE SQUARE OF A BINOMIAL (atb)² = (atb) (a+b) =a²+ab+ab+b²₂ Square of I [first term =a² + 2ab + b² Square of last term *Qx+3)(2x-3) is the product of the sum and difference of two kims Square of first AS Square the first term, Square the second term [+ Simplify Twice the product of the two terms (a-b)²-(a-b)(a-b)=a²-ab-ab+b² Lem Example 7-Multiply. Solution: (4c+56) ² -q²-2ab+b² Square of last term Twice the product of the two terms * (4c+ Sd)² is the square of a = (4c) ² + 2 (4c) (sd) + (sa)² _ binomial = 16c² + 40cd +25d²] Example 8-Multiply. Solution: (3x-2) = (3x)²-(2-3x) (2)+(2)² ) (2)+(2)² 1 = 9x² - 12x + 4) the product of the anating mas Square the terms Simplify Flex-2) is the square of a binomials. Square the terms and find the product of the two terms. B2 Simplify #13-41 3 x(x-2)=x²-2x 14 y(3-y)=3y-y² 13 -X(X+7)=-X¹²+(-7x) = -x ² - 7x (16) -y (7-y)= -7y-(-y²) - - 7y+y ² 17 3a²(a-2)-30³-60² 18 46² (b+8)=4b³+32b² @ 10 (28) 29 SxCx-x)= S(S) SAMSX3 20 -6y² (y + 2y²) = -6y³ - 12y"! (21-x² (3x²-7)=-3x² - (-7x²) == 3x² + 7x²³ (22-Y "Y" (2y² - y²) ==-2y² + y²⁰ (23) 2x (6x²-3x) =12x²³-6x² (29 3y (4y-Y²)=12y²-3y³ 63(2x-4)3x=6x²-12x 26 (2x+12x-4x²+2x (27) -xy(x² - y²) = no solution x²y (2xy-y²) = no solution x(2x²-3x+2)+2x²-3x² + 2x (30 y (-3y²-2y+6)= 3y ³ - 2y² + 6y 3-a-2a²-3a-2)-(20²³ +30² + 2a (32-b(Sb²+7b-35)=-5b³-7b²+35b 3 x²(3x²-3x²-2)=√3x3x-2x² 34 y ³ (-4y³ - 6y + 7) + 4√²-6√² + 7y³ 35 2y ² (-3y²-6y + 7) ==-6x ² - 12y²³ + 14x² x²(3x²-2x+6)=12x4-8x²³ +2²4x² (0²+3a-4) (-2a) = -2a²-69+80'l C (b³-2b+2)(-5b)--Sb² +10b²-106] -3y² (-2y²+y-2)=6y" - 3y² + 6y ² -SX²(3x²-3x-7) 15x³+15x² +35x²] *y (x²-3xy + y²¹ ¹X³²V-3x²³ y ²³²+xy²³²1 #45-59 (45), (Xx² + 3x + 2)(x+1) 36 37 •x²³ + 3x² + 2x + x² + 3x + 2 = √x ²³ + 4x² + 5x + 2] x-2x² + 7x-2x² + 4x-11-x² - 4x²+1|x - 14/ 19,(-26²-3b+4)(b-s) -2b²-3b²+ 4b+10b²+1Sb-20=26³-76² +19b-20 (48) (2x-3) (x²-3x+5) 2x² - 6x +10x+3x²+9x-15-2x²³-9x² + 12x-15 (a-3) (a²-3a+4) •9²³+30² +49-30²+9a-12-a³+13a-12) (-a²+3a-2) (2a-1) -2a²+a²+6a²-3a+Ya+2=20²³ +70²-7a+2] (S) (3x-s) (-2x²+7x-2) 4-6x³+2x²-6x+10x²-35x+10=E6x² + 3x²-41x + 10 X-²-2a+3) 1-2a²-4a²+6a-2a²+2a-3 = -2a²-6a²+8a-3 (x²-3x+2)(x-4) 4x²-3x²+2x - 4x³² + 12x-8-x-4x²-3x² +14x-8 (y ³ + 4y ²-8) (2y-7) 2 2 y ₁ + 8y³²-16y-y² - 4y² + 8 = 2y "-7y²³ - 4y² - 16y + 8 (SS), (3y-8) (³y² +8y-2) 30 (3 15y²³ +24y²-6y-40y²³-64-€25√²³ +24y²-6y-64 (4y-3) (3y² + 3y-5) 12v ₁²+12y ²³ - 20y=9y²-9y + 15 = 12√²+ 3y² = 29y + 15 67 (5a²³²-13a+2)(a-4) 45a-20a²³-15g²+60a+2a-8=Sa"-20a²-15a²+62a-8 (38) (36³-5b²+7) (6b-1) 18b4-3b²-30b²³ + 5b²+42b-7-18b+30b³ + 2b² +42b-7 (y+2) (Y³²+ 2y²-3y +1) 2 ²y² + 2y²³-3y² + y + 2√² + 4y²-6y + 2 = y² + 4y ²³+ y ²³ - 5y + 2² #67-101 (73)(y-7) (4-3) = y ₁²-3₁ -7x + 21 4√√²+²4y+21 (67) (x+1) (x+3) = x² + 3x + x + 3 = x² + 4x +3] + 3 = x ² + 4x +3] 68 (Y+2) (y+5)= y² + 5y + 2y + 10 = √² + 7y + 10 69 (a-3) (a+4)= a² +4a-3a-12-/a²-a-12 -_-a-12) 20 (b-6) (b+3)=b²+36-6b-18-16²-36-181 (79(a-8)(a-9 ) =a²-90-8₁ +72 •²+a+72 (y + 3), (y-8)= y² +94 + 3y - 24-1²-24 13(2x+1)(x + 7) = 2x² + 14x + √x + ²7 (x+10)(x-5)=x²-5x+10x-50= x² + 5x -50% 42x² + 15x+7 69 80 8 (82) 8 116 316 3 4880 101 22/0² 12 12 7 21 54 84 13 (4x-3 (16) (y+2) (Sy+1) = Sy²+y+/Jy+2=5y² +lly+2 (3x-1)(x+4)= 3x² + 12x-1x-4-3x² + 11x-41 (7x-2)(x+4)=7x²+28x-2x-8-7x²+26x-8) (4x-3)(x-7)=4x²+28x-3x+21=14x²+25x+21 (2x-3) (4x-7)=8x²-14x+12x+21=18x² + 2x + 21 3y-8)(y+2) 3y² +6y-8y-16-3y²-2y-161 (Sy-2)(y+5)= Sy², 25y-9y-45= 15y² +16y-45 (3x+7) (3x+11)= 9x² +33x+2/x+77-19x² + 4x + 77 (Sa+6) (60+5)=20a²+25a +12a+30=130a²+37a+30, (7a-16) (3a-5)=21a²-35a-48a+80=21a²-830+80 86 (Sa-12) (3a-7)=15a²-35a-36a+84-159²-71a +84] 87 (36+13) Sb-6)=156²-18h+65a-78=115b²+47a-78 (89(x+y)(2x+y) = 2x²+xy + 2xy + y² + y² + 2x² + 3xy 89 2 (2a+ba+30)=2a²+6ab+ba+3b²=3b²+2a² +Gab+ba/ 80 (3x-4) (x-2y)=3x² - 6xy-4yx+8y² =/8y²+3x² - 6xy-4yx (2a-b) (3a+2b)=6a²+ab-3ab-2bb-6a²+ab-2bb1 42 (Sa-36) (2a + 4b) = 10₂² +20ab-6ab-12b² = -12b²+10a² + 14ab 43(2x+y)(x-2y) = 2x² - 4xy + yx - 2y² = -²2y² + 2x² - 4xy+yx] (299) (3x - 7x) (3x + ³/2) = 9x² + 15xy = 21vx²-45v² = 145y² + 9x² +1Sxy-2/yx] Ⓒ (2x+3x) (5x + 7y) = 110x² +14xy + 15√x + 2y ² @6 (5x + 3y) (7x + 2y) = 35x ² + 10xy + 2/yx+6y² = √6y² + 35x² + Oxy + 2/yx (17 (3a-2b) (2a-76b)=6a²-2lab-4ba + 146²=114b² +69²-2/ab-4ba (5a-b) (7a-b)=35a²-Sab-7ba+b²=360²-Sab-7bal @9) (a-ab) (2a+7b) = 2a²+7ab-18ba-63b²=-63b²+2a²+7ab-18ba (2a + Sb) (7a-2b)=14a²-4ab+35ba-l0b² = 10b² + 14a²-4ab + 35ba (5x+2y) (2x-Sy)=10x²+25xy + 4yx-10y² = Eloy ² + 1 0x² - 2.5xy + 4yx] 65 7-8 #107-123 (odds) (107) (Y-S) (Y+S) = y² + Sy-Sy-25=y²-25₂ (09 (2x+3)(2x-3)=x²-6x + 6x-9= 4x²-9 (1)(x+1)^² = (x+2)(x+1)= X ² + 1x + 1x + 1 = X ² + 2x + 1] 13 (3a-5)=(3a-5) (3a-5)=9a²-15a-15a + 25 = 90² +251 (13 (3x-7)(3x+7)=x²+2/x-21x-49=19x²-49. (2a+b)²(2a+b) (2a+b) = 4a² + 2ab+ 2ab + b ² = 4a² + b ² ²² ² (119) (x-2y) ² = (x-2y) (x-2y)=x²-2√x-2yx + 4y² + x² + 4y ²₁ (2) (4-3) (4+3y) = 16+12y-12y-9/²= [6-9y²²] (2) (5x + 2y)² = (5x+2y) (5x+2y) = 25x² +10xy + 10xy + 4y ²= 25x² + 4x² J 11 #11-83 95-99 odds Section 74 Integer Exponents & Scientific Notation 1. The quotient of two exponential expressions with the same base can be simplified by writing each expression in fact pred form, dividing by the common factors, and then writing the XX.X.X.X.XX³ X² X.X 2. Note that subtracting the exponents results in the same quotient. bx². X² 3. To divide two monomials with the same base, subtract the exoments of the like bases Focus (Example)-Dividing exponential expressions IS Simplify. A. a³ B.r's 41054 8.6 5 8-7 6-1 rs =rs" 4. Recall that for any number a, a +0,&=1. This property is true for exponential expressions as well. Ex: for example, for x+O₁X² = 1 5. This expression also can be simplified using the rule for dividing exponential with the same base. =X²-4-X° ** The bases are the same. Subtract the exponent in the denominator from the exponent in the numerator. *Subtract the exponents of the like bases. 4-4. ZERO AS AN EXPONENT If x+0. then x 1. The expression O° is undefined. EXAMPLES: Simplify: (12²)", ato Any nonzero expression to the zero power is 1. 2 Simplify: -(¹) YO Any nonzero expression to the zero power is 1. Because the negative sign is outside the parentheses, the answer is-1. 6. The meaning of a negative exponent can be developed by examining the quotient x² وبلا (120³) = 1 7. The expression can be simplified by writing the numerator and denominator in factored form, dividing by the common factors, and then writing the result with an exponent. X² -(y¹) ⁰= -1 8. Now simplify the same expression by subtracting the exponents of the like bases. 4X²X²²X-² XY 9. Because and = x=², the expressions andx=² must be ea al. fending to the follow madefinition of a negative exponent DEFINITION OF NEGATIVE EXPONENTS Ifn is a positive integer and x/0, then x and = X". EXAMPLES: = -10 In each example below, simplify the expression by writing it with a positive exponent. 7a² Example-Evaluating a numerical expression with a negative exponent. Evaluate 2. 24 = 1 N Write the expression with a mative exprent. Simplity APULE FOR DIVIDING EXPOMENITAL EXPRESSIONS m-n If m and n are integers and x0, then EXAMPLES: Simplify each expression below by using the Rule for Dividing Exponential expressions. 0x²-x³-5-x-²-1 X5 6 +²2₂² = 16-(-2)=√ √ 8 Ⓒ Example 1-Write Solution: 33 --3 3² n =3-5 35 L » A₁ a ² b ² = b ² = A. la+ = 1 B.xy XY² 243 RULE FOR EXPONENTS If m, n, and pare integers, then: 0xm. -X²² m-n 3b-sb-³-(-¹)=√b²" √₁ = (by Lab³² Example 2-Simplity. BXY 0x²¹² „xy² b-1 with a positive exponent. Then evaluate. 23 and 13 hav *3 and 3" have the same base. Subtract the exponents 6 14 g 1-(+)_[ 10. An exponential expression is in simplest form when it is written with only positive exponents 93 * Use the Definition of Negative Exponents to write the expression with a positive & exponent *Evaluate.. Ⓒ (Xmyn)²=XmPynp 8x²7 ² 1² X+0 6X = 1, X 40 = *Use the Definition of Negative Exponents to rewrite the expression ©6d", do Flawale a with a positive exponent. 明 & Divide variables with the 5 X come base by subtracting the exponents. A Write the expression with only positive exponents. tr Example 3-Simplify →A-35a³b-² 6-2x)(3x²2)-3 250-265 -61-2 Solution: A.-35ab²_350³b-² A negative sign is placed 25a²b5 25a²b³ in front of a fraction = ²8.798-(-2) 1-2-3x Factor the coefficients. 8.5 =_7986²-79² 5 So (-2x)(3x-2)-³= (-2x) (3-³x) = -2x.x² 3³ 2x+ 27 Divide by the common factors. Divide variables with the same base by subtracting the same exmoments engineering. Example: The charge of an electron is coulomb. Write the expression with only Dositive exponents Use the rule for simplifying powers of products Write the expression with positive exmoments *Use the rule for multiplying exponential expressions, and simplify the numerical exponential. expression 11. Very large and very small numbers are encountered in the fields of science and 0.000000000000000000160 Can be written more easily in scientific notation. 12. In scientific notation, a number is expressed as a product of two factors, one a number between 1 and 10 and the other, a power of 10. 13. To change anumber written in decimal notation to scientific notation, write it in the form ax lor, where a is a number between 1 and 1.0 n is an Example: For numbers greater than 10, move the decimal paint to the right of the first digit. The exponentnis positive and equal to the number of places the decimal point has been moved. 240,000=2.4×105 93,000,000-9.3x10² Example: For numbers less than 1, move the decimal point to the right of the first nonzero digit. The exponent n is negative. The absolute value of the exponent is equal to the number of places the decimal point has been moved. 0.0003,0-3.0x10-4 0.0000832 = 8.32 x 10-5 (another example) - Using the definition of negative exponents, 10-5 = 1 1 100,000 Because 10-5=0.00001, we can write: 8.32 x 10 = 8.32 x 0.00001=0.0000832 Which is the number we started with. We have not changed the Love of 4- Wimber, we have just witten it in another form Example 4-Write the number in scientific notation. 824,300,000,000 0.000000961 Solution: A.824,300,000,000 -8.243 x 1011 10° 0.00001 B. 0.000000961 9.61 × 10-7 = Move the decimal, pant 11 places to the left. The exponent on 10 is 11 * Move the decimal paint 7 places to the right. The exponent on 10 is -7 14. Changing a number written in scientific notation to decimal notation • also requires moving the decimal point. Example: When the exponent on 10 is positive, move the decimal point to the right the same number of places as the exponent. 3.45 x 10 = 3,450,000,000 x 108 = 230,000,000 Example: When the exponent on 10 is negative, move the decimal point to the left the same number of places as the absolute value of the exponent. 8. 1x 10³ 0,0081 6.34 x 106 0₂00000634 Example 5-Write the number in decimal notation. A7329 x 10° B 6.8 x 10-10 Solution: A. 7.329 x 10°= 7,329,000 Ⓒ 2.3 B. 6.8 x 10-10=0.00000000068 The exponent on 10 is positive. Move the decimal point 6 places to the right. 15. The rules for multiplying and dividing with numbers in scientific notabon are the same as those for calculating with algebraic expressions. The power of 10 corresponds to the vanable, and the number between 1 and 10 corresponds to the coefficient of the variable. The exponent on 10 is negative. Move the decimal paint 10 places to the left Algebraic Expression Scientific Notation Multiplication (4x2x³) = 3x² Division 3x-2 (4x 10-3) (2 x 10) = 8x10² 6x2x-²)=2x²6x105-2x105-(-2) = 2x10² 3x102 Example 6-Multiply or Divide. A. (3.0x 10) (1.1 x 10^8) B. 7.2 x 10¹5 2.4 x 10-3 Solution: (3.0 x 10°) (1.1 x 10-8) 43.3 x 10-3 ® 7.2 x 10¹¹ - 3x 10²6 13 = 2.4 x 10-³ # 22₁29-2- Z2 1x1 *Multiply 3.0 and 1.1. Add the exponention 10 Divide 7.2 by 2.9. Subtract the exponents on 10 (13