Any combination of the prime factorization. MAT 120/121 Any combination of the prime factorization. Find the number that “GAZINTA” all the numbers. 6 goes into 12, 2 times and into 18, 3 times. The 6 and the 8 can still be divided by 2. There are no common factors in all the factors 4, 6, and 9. The GCF = 4 The only number that divides into 2 and 3 is 1…the number on the left side is the GCF. GCF = 6 The product of the numbers on the left side is the GCF. GCF = 4 * 2 = 8

Distributive Property. From the previous examples. Leftovers Leftovers Leftovers Distributive Property. When variables are a GCF, it will always be to the smallest power.

Factor each side by GCF’s MAT 120/121 Group the terms in half. Factor each side by GCF’s 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.

Factor by grouping

Make up any two binomials and FOIL them. MAT 120/121 F.O.I.L. 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 binomials. There is another pattern. Find the product of the F & L terms and O & I terms. Make up any two binomials and FOIL them.

This pattern gives us the a, b, c rule for finding our factors This pattern gives us the a, b, c rule for finding our factors. Number Sense! Factor by Grouping Factor the trinomials. Answer looks like. Number Sense Rules. Odd + Even = Odd Even + Even = Even Odd + Odd = Even 2 is factor every Even. (Odd)*(Odd) = Odd Big Answer looks like. 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!

Factor the trinomials. 5(-12) = -60 not 60! Prime

Prime doesn’t happen too often, so make sure you check everything! MAT 120/121 Prime doesn’t happen too often, so make sure you check everything! 1(-8) = -8 not 8! Prime

These directions means more than one factoring…Watch for GCF! GCF of -3x6 GCF of x2 GCF of 5 GCF of 4 GCF of -2 Can’t go any further because of the variables cubed. Not Prime…factored -2 out!

MAT 120/121 F.O.I.L. 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 binomial are and FOILing them out to test if the middle term is correct. 8 tries to get the right answer!?!

MAT 120/121 Refers to the middle term. ODD + EVEN = ODD 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. odd even 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! Do you see that? Say YES! What terms are generating these binomials? Look above each step. Answer looks like. 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. It should still factor if we switch the 15x and -8x. Here is a shortcut. Always put the “a” in both binomials. Put in the factors. 2 5 Take out GCF’s

Refers to the middle term. EVEN + EVEN = EVEN 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. 2( ) 2( ) Factor 2 out of the -14. The sum of the two red ( )’s must = – 7. even even Because a = 8 Since – 7 is odd. Isolate the odd factors and multiply all possible odd combinations. 6 and -20 are the two factors that add up to -14. Place them in our answer. Answer looks like this using new short cut. Use 8x twice. 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. 2 4

Because both a & c are odd EVEN RULE ( odd + odd ) Refers to the middle term. ODD + ODD = EVEN Factor. 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. 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. Wrong factors! Right factors! Answer looks like this using new short cut. Use 3x twice. 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. NO GCF 3

These directions means more than one factoring…Watch for GCF! MAT 120/121 These directions means more than one factoring…Watch for GCF! The 2 or 4 must be multiplied to the 3 The 5 and -6 doesn’t work, so try 3 and 10! -3 and 10 work. even even Because c = -8 odd even NO GCF 3 5 NO GCF Remember this example 2 pages ago, where the author FOILed it out 8 times? Which way is easier? GCF of 2x. One of the 2’s must be multiplied to the 5. 2 and 10 Because 11 is much bigger than 4 and 3, multiply 4 and 3 to get 12. even even Because a = 4 odd even 2 2 4 NO GCF

Factor completely. PRIME 10 4 2 3 2 63 is a big value… factors must be far apart. Need to have x powers in descending order. odd even odd odd even odd No possible factors. PRIME NO GCF 10 One of the 3’s must be mult. to the 2, 6 and 3 subtract to be the 3 in the ( )’s. 3*_____ 5*_____ 15*____ 40 2( ) 2( ) 24 even even even odd even odd 8 4 2 3 2

Factor completely. GCF of -3. GCF of 2x2. 3 2 2( ) 2( ) 6 2 6 3*_____ 5*_____ 9*____ 75 45 odd even odd odd odd even 25 3 2 The two 3’s mult. together, 9 and 5 add up to be the -14 in the ( )’s. 3*_____ 7*_____ 21*____ 56 2( ) 2( ) 24 odd even odd even even even 8 NO GCF 6 2 6

We must test the middle term! It Factors! MAT 120/121 It is important to know that x2, x4, xeven, etc. are all perfect squares Per. Sqr. Per. Sqr. Let’s try a ( )2 We must test the middle term! It Factors!

Done! Done! Done! NOT Done! ERASE Done! Done! Done! 4 and 36 are perfect squares, but 4 is a GCF! Done! Done! Done!

HIDE inside other polynomials! MAT 120/121 HIDE inside other polynomials! GCF of 9, 1st!

Factor completely. NOT Diff. of Per. SQ! GCF of 3, 1st! Prime Another Diff. of Per. SQ! Or put the 25y2 in front. GCF of -1, 1st! Even powers on the variables are still perfect squares.

MAT 120/121 Middle terms exist! Binomial Trinomial Same sign as given Opposite sign as given Always Plus

1st Both Diff. of Per. Sq. and Perfect Cubes 2nd

MAT 120/121 GCF( LEFTOVERS ) 1st 2nd The number of terms in the leftovers determines which step we go to. 1st Remember these like to hide inside of other polynomials. 2nd Same sign as given Opposite sign as given Always Plus  

Difference of Perfect Squares Remember the sign rules for what your answer looks like. GCF GCF 3 to 1 SPLIT 1 to 3 SPLIT is the same concept, but watch for signs! GCF LT GCF RT Difference of Perfect Squares SAME SAME GCF LT GCF RT SAME

MAT 120/121 GCF of 5 GCF of 2x Step 4 F. by G. Step 2 D.P.S. Step 2 Again Step 2 D.P.S. Step 2 D.P.S. GCF of 3 Step 2 & P.C. twice Step 4 F. by G. Step 2 D.P.S. & P.C.

Factor completely. GCF of 7 GCF of 3x2 Step 3 Step 3 3 to 1 SPLIT F. by G. Step 4 F. by G. GCF of -1 first. Difference of Per. Squares Difference of Per. Squares Distribute the minus!

MAT 120/121 if (A)(B) = Remember the product of -8(10)(3)(5)(0)(7)(11) = 0. Solve each for x.

Solve the equations by factoring. Never divide by the variable! Set the equation = 0. We don’t have to list the four twice, but just know that there were two answers that were the same value. No reason to work out the 2nd binomial because the only difference will be the sign.

The factors have to differ by 1, so 2(7)=14 and 3(5)=15 Solve the equations by factoring. The factors have to differ by 1, so 2(7)=14 and 3(5)=15 odd even odd 2 3

Solve the equations by factoring. One of the 3’s must be isolated, 3 and 18 will subtract to be 15 in the ( )’s. 2( ) 2( ) even even even To save time we can solve these binomial right now. You will have to reduce.

Solve the equations by factoring. Will need to FOIL and set = 0 odd even odd

The dimensions are 40 cm by 20 cm. MAT 120/121 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. We know that the Length is twice the Width. The dimensions are 40 cm by 20 cm.

MAT 120/121 Multiply the two numbers and set = 156 Means that the numbers differ by 1. The First number is unknown call it x. The Second number must be 1 bigger… x + 1 Assuming the racing number must be positive, the first number is 12 and the consecutive second number is 13. There are two sets of answers! Means that the numbers differ by 2. The First number is unknown call it x. The Second number must be 2 bigger… x + 2 -22 and -20 20 and 22

The other two sides are 9 ft and 12 ft. Right triangles have a special relationship called The Pythagorean Theorem. 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. The other two sides are 9 ft and 12 ft.

There will be 210 micrograms in the bloodstream at 3 minutes and 7 minutes.

The minimum length of the cable is 125 ft. The two distances are 30 ft. and 40 ft.

A number is 6 less than its square. Find all such numbers.