Factor completely X^2-5x-24

Question

1.)  X^2-5x-24

2.)  3x^3-5x^2+2x

3.)  6ax-8ay+9bx-12by

Answer 1

1). X^2-5x-24

       1. Find the product of first term (1) and last term (-24) =-24 which add up to the middle term (-5); we'll  try a few factors which equal -24 but add up to -5 --           will 4*-6 work?                                                                                                                                         No: because 4+-6 = -2                                                                                     How about 3 * - 8?                                                                                         yes: Because 3+ -8 = -5  

        2.      X^2-5x-24 =   X^2 + 3x - 8x - 24           

        3.      Factor out the GCF from each side in:     x^2 + 3x - 8x - 24                                                           x(x + 3) (- 8x - 24)                                              (Think: what can you divide out of (-8x - 24) to get (X + 3)?)                                       Answer: -8, because -8 is distributed to (x +3) we get (- 8x - 24),                          so we have x(x + 3) -8 (x + 3) which = (x + 3) (x - 8)

       4. Solution is (x + 3) (x - 8)

 

2.)  3x^3-5x^2+2x

      1. Identify the GCF  3x^3-5x^2+2x; it is "x"

      2. Divide the GCF out of every term in the polynomial: x (3x^2 - 5x + 2)

      3. Find the product of  first term (3) and last term (2) =6 which add up to the middle term (5); we'll use what's given to us in this case, which is 3 and 2:                              x (3x^2 - 5x + 2)                                                                                            x (3x^2 - 3x - 2x + 2)                                                                                      Because -3*-2= 6, and when combined = 5

      4. Factor out the GCF from each side in  (3x^2 - 3x - 2x + 2)                                           (3x^2 - 3x) (- 2x + 2)                                                                                       3x (x - 1) (- 2x + 2)                                                                       (Think: what can you divide out of (-2x +2) to get (x - 1)?)                                         Answer: -2, because  when -2 is distributed to ( x - 1) we get: (- 2x + 2)                                    so, we have 3x (x - 1) -2 (x - 1) which = (3x - 2) (x - 1)

      5.  the solution is:  x (3x - 2) (x - 1)

                    

       

3.)  6ax-8ay+9bx-12by

      1. Divide out  the GCF of each side in 6ax-8ay+9bx-12by                                                                                 Use parenthesis to help with the process  (6ax-8ay) + (9bx-12by)                                                                 2a (3x - 4y)  +  (9bx-12by)                                                                                                                        (Think: what can you divide out of (9bx-12by) to get (3x - 4y)?)                                                                  Answer: 3b, because when 3b is distributed to (3x - 4y) we get: (9bx-12by)                                                so, we have 2a (3x - 4y) + 3b (3x - 4y) which = (2a + 3b) (3x - 4y)                                                                    

     2. The solution is (2a + 3b) (3x - 4y) 

Answer 2

1)  (x+3)(x-8)

2) x(x-1)(3x-2)

3) x(6a+9b) - y (8a-12b) - OR a(6x-8y) + b (9x-12y), depends on if you want to facter out the a/bs or the x/ys.

Answer 3

Look at the factors of your free term in the first equation: 24 = 1x24 = 2x12=8x3=6x4. Only 8 and 3 can make

-5: 3-8 = -5. Because the first coefficient is 1, this means your factors are (x +3)(x-8)

2) "x" is a common factors for all terms. Take it out of parenthesis and write:

           3x³ -5x² +2x = x(3x² -5x +2)

Now you have to break the quadratic trinomial into a  product of two , (3x-1)(linear binomials. The first coefficient is 3, and the last term is 2. Both are prime numbers with fcators 3 and 1 and 2 and 1 corrspondingly. Because 2 has plus sign thelast terms in the binomial factors should have the same sign. Possible options include

                        (x+1)(3x+2),   (3x+1(x+2),   (x-1)(3x-2),     (3x-1)( x-2)

but only one (x-1)(3z-2) can provide -5x for the second term in the quadratic expression. Thus, you will obtain

                                             3x³ -5x² +2x = x(x-1)(3x-2)    

3)  6ax - 8ay + 9bx -12 by = (6ax + 9bx) - (8ay +12 by) = 3x(2a+3b) - 4y(2a+3b) = (3x-4y)(2a+3b)