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2x 6x2 12 x x 2 4 x 3x 2 Any combination of the prime factorization. 2 2 3 x x x Find the number that “GAZINTA” all the numbers. 6) 2 3 6 goes into 12, 2 times and into 18, 3 times. The only number that divides into 2 and 3 is 1…the number on the left side is the GCF. GCF = 6 4) 2) 6 3 4) 8 4 The 6 and the 8 can still be divided by 2. The product of the numbers on the left side is the GCF. GCF = 4 * 2 = 8 4 6 9 There are no common factors in the remaining factors of 4, 6, and 9. The GCF = 4 From the previous examples. 6 2x 6 3 6 2 x 3 8 3a 8 4b 4 4x2 4 6x 4 9 8 3a 4b Leftovers 4 4x2 6x 9 Leftovers Leftovers Distributive Property. When variables are a GCF, it will always be to the smallest power. 3 4x 3 1 34x 1 4ab a 4ab 10b 4aba 10b 3x 2 y 2 x 3 5 xy 2xy3 3y 2 6x 2 x 3 4 x 3 x y 2a 3b Group the terms in half. Factor each side by GCF’s 2 x 1 1 5x ___ 3 ___ 5x ___ ______ ___ Since the 1st two terms are subtracting, both ( )’s will have minus signs. In fact, both sets of ( )’s must be the same for this to factor. If not the same, prime. Factor the same binomials as a GCF. 5x 1__x 2 __3 Factor by grouping 2 x3 8x 2 x 4 2 2x x ___ x ___ 4 ___ 4 1 ___ ______ x 42x __2 __ 1 8 x 4 6 x 28 x 3 21 2 x ___ 7 ___ 3 3 4x 3 ___ 4x 3 ___ ___ ___ 4x 3 7 x __ 3 2__ 6ax 3bx 4ay 2by 6w 10wx 35x 21 3 x ___ 2 y ___ b ___ b 2 a ___ ___ 2 a ___ 2 w ___ 7 ___ 5x 3 ___ ___ ___ 5x 3 ___ 2a b3__x __ 2 y 5x 32__w __7 F.O.I.L. F L 10x 2 O I 10x 2 When the “c” term is positive it means that the binomials have the same signs, and the sign on the “b” term determines the signs of the binomials. F L 10x 2 O I 10x 2 F L 10x 2 O I 10x 2 F L 10x 2 O I 10x 2 When the “c” term is negative it means that the binomials have the opposite signs, and the sign on the “b” term determines the signs of the largest value in the binomials. There is another pattern. Find the product of the F & L terms and O & I terms. Make up any two binomials, with no GCF, and FOIL them. This pattern gives us the a, b, c rule for finding our factors. ax 2 bx c I ac O ___ ___ ___ b I x bx O x ___ ax 2 Ox Ix c ______ ___ ______ ___ Factor by Grouping Factor the trinomials. x 2 8 x 12 Number Sense! x 2 8 x 20 x 2 5x 6 ___ 2 a c 1 12 6 ___ 2 b 8 ___ 6___ ___ ___ 2 a c 1 20 10 2 b 8 ___ ___ 10 ___ 2 a c 1 6 3 ___ 2 b 5 ___ 3 ___ x 2 6 x 2 x 12 x 2 10 x 2 x 20 3 __ 2 __ x __ x __ 2 ___ 6 ___ 6 x ___ x ___ x ___ ___ 2 6 __ x __ x __ __ Answer looks like. 2 ___ 10 ___ 10 x ___ x ___ x ___ ___ x 10 x __ 2 __ __ __ Big Answer looks like. Did you just notice that the numbers in the binomial answers are the same numbers that were our factors? This will always happen when a = 1! Answer looks like. Number Sense Rules. Odd + Even = Odd Even + Even = Even Odd + Odd = Even 2 is a factor every Even. (Odd)*(Odd) = Odd Factor the trinomials. 1 a c 1 6 ___ 6 ___ 1 b 5 ___ 6 ___ even + odd odd 1 6 x __ x __ ___ ___ 12 c 24 even 2 2 ___ 12 b 10 ___ even even + even 2 x 12 x __ __ ___ 5 ___ 15 c 75 odd 5 ___ 15 b 20 ___ even odd + odd 5 x 15 x __ __ (odd)(even)=even (odd)*(odd) = odd ___ 3 ___ 7 a c 1 21 ___ ___ 3 7 b 4 even odd + odd 3 x __ 7 x __ ___ ___ 4 6 a c 1 24 6 b 10 ___ 4 ___ even even + even 4 x __ 6 x __ ___ 8 c 24 3 ___ ___ 3 ___ 8 b 5 odd odd + even 3 ___ ___ 8 c 24 ___ 3 ___ 8 b 11 odd odd + even 8 3 x __ x __ 8 3 x __ x __ ___ 12 c 48 even 4___ 12 b 8 4 ___ ___ even even + even 12 4 x __ x __ 12 c 60 ___ 5 ___ 12 b 7 5 ___ ___ 5(-12) = -60 not 60! x __ x __ Prime Prime doesn’t happen too often, so make sure you check everything! ___ 2 ___ 18 c 36 even 18 b 20 ___ 2 ___ even even + even ___ 4 ___ 5 c 20 5 b 1 4 ___ ___ ___ 6 c 6 1 ___ ___ 1 ___ 6 b 7 2 x 18 x __ __ 5 4 x __ x __ 6 1 x __ x __ 1 ___ ___ 8 c 8 8 b 7 1 ___ ___ 1(-8) = -8 not 8! x __ x __ Prime 3 c 45 odd ___ 15 ___ ___ 15 ___ 3 b 12 odd + odd even 15y x __3 y x __ 8 c 24 ___ 3 ___ ___ 3 ___ 8 b 5 8 n m __3 nm __ These directions means more than one factoring…Watch for GCF! GCF of 5 5 x2 x 6 GCF of x2 2 3 c 6 ___ 2 ___ 3 b 1 ___ 2 ___ 12 x 11 x 2 x __ __ 2 x 6 x 18 2 ? ___ ? c 18 even ___ ? ___ ? b 6 ___ even even + even 2x __ x __ ___ 11 c 132 12 ___ 11 b 1 ___ ___ 12 3 2 x __ 5x __ GCF of -2 x x 2 x 132 GCF of -3x6 2 x 2 6 x 18 Not Prime…factored -2 out! GCF of 4 3x 6 x 2 4 x 5 5 c 5 ___ 1 ___ 5 b 4 1 ___ ___ 5 1 x __ 3x 6 x __ 4 x3 2 x 2 y 2 12 y 3 Can’t go any further because of the variables cubed. F.O.I.L. F L 60x 2 O I 60x 2 F L 60x 2 O I 60x 2 F L 60x 2 O I 60x 2 F L 60x 2 O I 60x 2 When the “c” term is positive it means that the binomials have the same signs, and the sign on the “b” term determines the signs of the binomials. When the “c” term is negative it means that the binomials have the opposite signs, and the sign on the “b” term determines the signs of the largest value in the FACTORS not the binomials. The author actually suggested guessing what the binomials are and FOILing them out to test if the middle term is correct. 8 tries to get the right answer!?! Refers to the middle term. ODD + EVEN = ODD 2 5 3 4 ___ 15 ___ 8 a c 10 12 ___ 15 ___ 8 b 7 odd even 10 x 2 15 x 8 x 12 4 ___ 5x ___ 2x ___ 3 ___ 2x ___ 3 ___ 2x __ 3 __ 5x 4__ __ Answer looks like. It should still factor if we switch the 15x and -8x. 10 x 2 8 x 15 x 12 3 ___ 2x ___ 5x ___ 5x ___ 4 ___ 4 ___ 3 2x __ 4 __ 5__x __ Since we have an odd + even, we need odd factors. Break the 10 and 12 down to odd factors. Isolate the odd factors and multiply all possible odd combinations. 3 ___ 40 5 ___ 24 15 ___ 8 3 5 2 4 3 40 7 5 3 2 4 5 24 7 5 3 2 4 15 8 7 Not the factors Not the factors Right factors I can see a pattern! When you look at the left side of each factoring by grouping, I see the two binomials in the answer! Do you see that? Say YES! What terms are generating these binomials? Look above each step. It is the leading term and the two factors! Can we all agree that we will always factor out at least an x as the GCF? Yep. Here is a shortcut. Always put the “a” in both binomials. Put in the factors. Take out GCF’s 8 10x __ 15 10x __ 2 5 5x 42x 3 Refers to the middle term. EVEN + EVEN = EVEN 222 3 5 Since we have an even + even, we factor out a 2 from our factors. Break the 8 down to get factors of 2’s. Put a 2( ) in each blank as a factor because we know that the two factors are even. 20 a c 8 15 6 _____ _____ 2( _____ 3 ) 2(_____ 10 ) b 14 2 ( – 7 ) Factor 2 out of the -14. The sum of the two red ( )’s must = (– 7). even even Because a = 8 6 and -20 are the two factors that add up to -14. Place them in our answer. Answer looks like this using the new short cut. Use 8x twice. 20 6 8x __ 8x __ 2 4 4x 32x 5 Since – 7 is odd. Isolate the odd factors and multiply all possible odd combinations. 3 ___ 10 3 2 5 3 10 7 5 ___ 15 ___ Right factors! Put them in the red ( )’s! Now we know we are not finished because we used the 8 twice. We have to divide out the extra 8 by finding the GCF of each binomial. Refers to the middle term. ODD + ODD = EVEN EVEN RULE ( odd + odd ) Factor. 3x 2 34 x 63 337 7 _____ 27 a c 3 63 _____ _____ 7 _____ 27 b 34 odd odd Because both a & c are odd -7 and -27 are the two factors that add up to -34. Place them in our answer. Since a and c as odd factors we have an odd + odd = -34. This is going to take some time because all the factors will be odd. Break 63 down. Isolate each odd factor, from smallest to largest, and then multiply all possible odd combinations to create more odd factors. 3 ___ 7 ___ 9 ___ 63 27 3 3 3 7 7 3 3 3 3 63 34 7 27 34 Wrong factors! 21 ___ 27 ___ Right factors! Answer looks like this using new short cut. Use 3x twice. 7 3x 27 3x __ __ NO GCF 3 3x 7x 9 Now we know we are not finished because we used the 3 twice. We have to divide out the extra 3 by finding the GCF of each binomial. These directions means more than one factoring…Watch for GCF! 2 3 2 4 The 2 or 4 must _____ 2 _____ 12 a c 3 8 2 _____ 12 b 10 _____ be multiplied to the 3 even even Because c = -8 odd 2 3x 12 3x __ __ NO GCF GCF of 2x. 3 3x 2x 4 Remember this example 2 pages ago, where the author FOILed it out 8 times? Which way is easier? 2 x 4 x 11x 3 2 Because 11 is much bigger _____ 1 _____ 12 a c 4 3 than 4 and 3, multiply 4 and 3 1 _____ 12 b 11 _____ to get 12. odd even 2x4x 12 __ 4x __ 1 4 NO GCF _____ 3 _____ 10 a c 5 6 3 _____ 10 b 7 _____ 2 xx 3 4x 1 The 5 and -6 doesn’t work, so try 3 and 10! -3 and 10 work. even 3 5x 10 __ 5x __ x 25x 3 NO GCF 5 22 One of the 2’s must _____ 10 a c 4 5 be multiplied to the 2 _____ 10 b 12 5. 2 and 10 2 _____ _____ even even Because a = 4 2 4 x __ 10 4x __ 2 2 2 x 12x 5 Factor completely. Need to have x powers in descending order. 6 x 2 4 y 2 19 xy 6 x 2 19 xy 4 y 2 63 is a big value… factors must be far apart. 10 x 2 63x 18 2 3 _____ 3 _____ 60 a c 10 18 3 _____ 60 b 63 _____ 1 24 _____ _____ a c 6 4 3 8 _____ _____ b 19 No possible factors. odd even odd PRIME 6x __ y 6x __ y 6 x 23x 20 2 _____ a c 6 20 15 _____ 8 _____ b 23 15 _____ 8 odd even 8 6x 15 __ 6 x __ 3 2 odd even odd 40 3*_____ 24 5*_____ 15*____ 8 2x 5 3x 4 odd 60 3*_____ 5*_____ 9* ____ 15*____ 45*____ 3 10x __ 60 10 x 3x 6 10x __ NO GCF 2 3 225 2 5 233 10 222 One of the 3’s 3 3 must be mult. to 8x 6 x 9 6 _____ 12 a c 8 9 the 2, 6 and 3 _____ subtract to be _____ _____ b 6 3 6 23 the 3 in the ( )’s. 2( ) 2( ) 2 even even 6 8x 12 __ 8x __ 4 2 even 2x 3 4x 3 Factor completely. GCF 30 x 4 28 x 3 30 x 2 of 2x2. 2 x 15 x 14 x 15 2 2 GCF 2 of -3. 18 x 39 x 18 3 5 3 5 3 6 x 13x 6 _____ _____ a c 15 15 75 3*_____ _____ _____ b 14 odd odd 2 x 2 15 x __ 15x __ 2 x 15 x 14 x 15 2 2 6 x 2 17 xy 28 y 2 even 45 5*_____ 9*____ 25 even 7 y 6x 24 __ y 6x __ 6 NO GCF x 4 y 6x 7 y odd odd even 24 7*_____ 21*____ 8 odd 9 6x __ 4 36x __ 2 32 x 3 3x 2 2 3 227 3 2 3 2 _____ 9 _____ 4 ac 66 4 b 13 _____ 9 _____ 3 _____ 7 _____ 24 a c 6 28 56 3*_____ 24 b 17 _____ 7 _____ odd 2 223 12 x 2 28 x 15 The two 3’s mult. together, 9 and 5 add up to be the -14 in the ( )’s. 3 5 _____ 18 _____ 10 a c 12 15 2( 9) _____ 2( 5) b 28 214 _____ even even even 12x 18 __ 12 x 10 __ 6 2 2x 3 6x 5 It is important to know that x2, x4, xeven, etc. are all perfect squares Per. Sqr. x2 4x 4 x 2 2 Per. Sqr. Let’s try a ( )2 We must test the middle term! 2x 2x 4x It Factors! x 52 x 4 2 Done! 5x 5x 10x x x__13x2 __ 9 Done! 4x 4x 8x Done! NOT Done! 3x 3x 6x ERASE 4 and 36 are perfect squares, but 4 is a GCF! 4 x 6x 9 2 4x 3 2 Done! 3x 3x 6x 4 x 3 3x 7 2 Done! 12x 12x 24x 2 Done! 21x 21x 42x HIDE inside other polynomials! x 5x 5 x 8x 8 GCF of 9, 1st! 9 x2 4 9x 2x 2 Factor completely. 16 x 4 81 4x 4x 2 2 9 4x2 9 x2 4 12 x 2 75 NOT Diff. of Per. SQ! PRIME Another Diff. of Per. SQ! 9 2 x 32 x 3 GCF of 3, 1st! 3 4 x 2 25 32x 52x 5 Or put the 25y2 in front. 36 x 2 25 y 2 GCF of -1, 1st! 1 36 x 25 y 2 2 25 y 2 36 x 2 16 x 5 y 6 x 5 y 5 y 6x5 y 6x 100a 4c 6 121x8 y 2 Even powers on the variables are still perfect squares. Divide the powers by 2 to take the square root. 10a c 2 3 11x 4 y 10a 2c3 11x 4 y Middle terms exist! Binomial 3 Trinomial Same sign Opposite Always Plus as given sign as given b ___ a ___ ab ___ a 2 ___ b2 a b ___ 3 3 3 3 3 3 y __ 2x____ ___ 4x 2 9 y 2 ____ 6 xy ____ 20 x ____ 16x 2 ____ 25 4x ___ 5 ____ ___ 3 3 3 2 4 2 2 36 x y 6 x y ____ 6 x y __ ____ ____ ___ 2a 3 8x3 3 t 3 3 3 3 2 70 x ____ 49 x ___ 7 100x 10 ___ ____ ____ 1st Both Diff. of Per. Sq. and Perfect Cubes 2nd 3 8x 1 8x3 1 33 t ___ 2a___ __ 4x 2 2___ xt ___ t2 2x 1 ___ 2 x __ 1 ___ 1 x __ 4x 2 ___ 2__x __ 1 2__ 4x 2 ___ 2 x __ GCF( LEFTOVERS ) The number of terms in the leftovers determines which step we go to. a 2 b 2 a ba b 1st Remember these like to hide inside of other polynomials. 2nd 3 Same sign Opposite Always Plus as given sign as given b ___ a ___ ab ___ a 2 ___ b2 a b ___ 3 3 3 Remember the sign rules for what your answer looks like. e ac d _____ _____ d _____ e b _____ d ax __ ax __ e GCF GCF 3 to 1 SPLIT a 2 x 2 2abx b 2 y 2 ax 3 bx 2 cx d ______ ___ ___ RT ___ ___ GCF LT GCF SAME SAME ___ ___ ___ ___ GCF LT SAME GCF RT ax b 2 y 2 Difference of Perfect Squares 1 to 3 SPLIT is the same concept, but watch for signs! GCF of 5 Step 2 D.P.S. Step 2 D.P.S. Again 5 x 16 4 GCF of 2x 5 x 4 x 4 2 2 2 xx 2 x 5 92 x 5 2 x2 x 5x 9 2 x 2 x 3 5 x 2 18x 45 Step 4 F. by G. 2 2 5 x 2 4 x 2x 2 Step 2 D.P.S. 2x2x 5x 3x 3 Step 2 D.P.S. x 8x 8 Step 2 Step 4 & P.C. F. by G. 3x x 1 8x 1 twice x 2x 2 x 4x 2x 3x 1x 8 Step 2 D.P.S. 3x 1x 1x 2x 2 x 4 & P.C. GCF of 3 3 x5 x3 8x 2 8 3 2 3 2 2 2 3 3 2 2 2x 4 Factor completely. GCF of 7 Step 3 7 x 5x 6 2 GCF of 3x2 7x 3x 2 Step 3 3 to 1 SPLIT Step 4 F. by G. x 3 2 y Difference of Per. Squares 3x 2 x 2 10 x 24 3x 2 x 12x 2 1 to 3 SPLIT Step 4 F. by G. 2 x 3 y x 3 y x 3 yx 3 y y x 6x 9 2 y 2 x 3 2 2 GCF of -1 first. Difference of Per. Squares y x 3 y x 3 Distribute the minus! y x 3 y x 3 Remember the product of -8(10)(3)(5)(0)(7)(11) = 0. 5x 0 x2 0 2 2 Solve 5 5 each x0 x2 for x. x 0,2 x 4x 6 0 x4 0 x4 7 0 x 8 0 2x 3 0 x6 0 x 4,6 x6 x 8 3 x 8, 2 2x 3 3 x 2 2x 0 x 1 0 x 1 0 x0 x 1 x 1 x 0,1,1 x 12x 2 0 x 12 x 2 x 12,2 Solve the equations by factoring. x 2 8 x 16 0 x 4x 4 0 x4 We don’t have to list the same number twice, but just know that there were two answers that were the same value. Never divide by the variable! Set the equation = 0. x2 6x 4 x 2 25 x2 6x 0 xx 6 0 4 x 2 25 0 x0 x6 x 0,6 2x 52x 5 0 2x 5 0 2x 5 5 x 2 5 x 2 5 5 x , 2 2 No reason to work out the 2nd binomial because the only difference will be the sign. Solve the equations by factoring. x 2 16 x 63 0 x 7x 9 0 x 7 x 9 x 7 , 9 2 2 x x 2 4 x 12 0 2xx 2x 6 0 x0 x2 x 6 x 0,2,6 x 14 x 2 x 14,2 2 x 8 x 24 x 0 3 x 2 12 x 28 x 2 12 x 28 0 x 14x 2 0 6 x x 35 0 2 2 3 75 _____ 15 _____ 14 a c 6 35 _____ 15 _____ 14 b 1 odd even odd 6x 14 __ 6x 15 __ 0 2 3 The factors have to differ by 1, so 2(7)=14 and 3(5)=15 3x 72x 5 0 3x 7 0 3x 7 2x 5 0 2x 5 7 x 3 5 x 2 Solve the equations by factoring. One of the must be 8 x 2 30 x 27 0 2 2 2 3 3 33’s isolated, 3 6 a c 8 27 and 18 will 36 _____ _____ subtract to 2( ) 2( ) _____ 18 _____ 3 b 30 2 15 be 15 in the even even even ( )’s. 368x __ 6 0 8x __ 4 2 2x 9 0 4x 3 0 2 x 9 9 x 2 4x 3 3 x 4 9 3 x , 2 4 2 x 3 3x 2 8 x 12 0 x 2 2 x 3 42 x 3 0 2 x 3x 2 4 0 2x 3x 2x 2 0 2x 3 0 x 2 2x 3 3 x 2 3 x 2,2, 2 x2 Solve the equations by factoring. x 32x 1 9 Will need to FOIL and set = 0 2x2 x 6x 3 9 2 x 2 5 x 12 0 223 _____ 3 _____ 8 a c 2 12 _____ 3 _____ 8 b5 odd even odd 8 2x __ 3 0 2x __ x 3x 4 5x x 2 4 x 3x 12 5 x x 2 1x 12 5 x x 2 4 x 12 0 x 6x 2 0 x 6 x 2 2 x40 x 4 2x 3 0 2x 3 3 x 2 3 x , 4 2 x 6,2 The cutting board is a rectangle because of the reference to “long and wide.” Build a rectangle. The Area formula is L * W and the Area equals 800. W L W Area L 2W 2W W 800 We know that the Length is twice the Width. 2W 2 800 2W 2 800 0 2 W 2 400 0 2W 20W 20 0 W 20 W 20 L 2W L 2 20 40 The dimensions are 40 cm by 20 cm. Multiply the two numbers and set = 156 Means that the numbers differ by 1. The First number is unknown, call it x. x x 1 156 The Second number must be 1 bigger… x + 1 x 2 1x 156 Assuming the racing number must be positive, the first number is 12 and the consecutive second number is 13. x 2 1x 156 0 x 13x 12 0 x 13 x 12 Means that the numbers differ by 2. The First number is unknown call it x. The Second number must be 2 bigger… x + 2 x x 2 440 x 2 2 x 440 There are two sets of answers! x 2 2 x 440 0 x 22x 20 0 x 22 x 20 -22 and -20 20 and 22 c a a b c The legs of the right triangle are the sides of the right angle, labeled a and b. The hypotenuse is the longest side and is labeled c. b x 9 Right triangles have a special relationship called The Pythagorean Theorem. 2 2 2 15 a2 b2 c2 x 2 x 3 152 2 x3 9 3 12 The other two sides are 9 ft and 12 ft. x 2 x 2 3x 3x 9 225 2 x 2 6 x 216 0 2 x 2 3x 108 0 2x 12x 9 0 x 12 x 9 N 210 10t 2 100t 210 10t 100t 210 0 10 t 2 10t 21 0 2 10t 3t 7 0 t 3 t 7 There will be 210 micrograms in the bloodstream at 3 minutes and 7 minutes. 135 352 120 2 c 2 1225 14400 c 2 c 35 0 c 125c 125 15625 c 2 100 120 0 c 2 15625 c 125 The minimum length of the cable is 125 ft. h 2 h 10 50 2 2 50 h 10 h 2 h 2 10h 10h 100 2500 h 2h 2 20h 2400 0 2 h 2 10h 1200 0 2h 40h 30 0 h 30 The two distances are 30 ft. and 40 ft. A number is 6 less than its square. Find all such numbers. x 2 x 6 0 x2 x 6 0 x 2x 3 x 2 x 3 x x2 6 2 2 2 6 2 46 x x2 6 3 3 6 3 96 2