how to factor each of these polynomials step by step

Question

x²+14x+24

 

2x-7x-15

 

x²-6x+9

 

x²-144

 

x²-25

 

2x²+7x+3

Answer 1

x² + 14x + 24

x² + 12x + 2x + 24

x(x+12) + 2(x+12)

(x+2)(x+12)

 

2x- 7x -15

-5x - 15

-5(x+3)

 

x² - 6x +9

x² -3x -3x +9

x(x-3) -3(x-3)

(x-3)(x-3)

 

x² -144

x² - 12²

(x-12)(x+12)

 

x² -25

x² - 5²

(x-5)(x+5)

 

2x² +7x +3

2x² 6x + x + 3

x(x+3) +(x+3)

(x+1)(x+3)

:)

Answer 2

Britt,

 

It is a "trial & error" process to find the right factors, but it is guided by logic to reduce the number of needed trials.  To see this, let's start out with the most general two factors for any quadratic polynomial:

 

(ax + b)(cx + d) = (ac)x² + (bc + ad)x + (bd),

 

 

where a, b, c, d can be negative, zero or positive and of different magnitude individually.  But notice their relationships that you must keep in mind when you do factoring:

 

a and c multiplied together must produce the sign and magnitude of the x² term in your polynomial.

b and d multiplied together must produce the sign and magnitude of the constant term in your polynomial.

bc + ad  must produce the sign and magnitude of the middle x term in your polynomial.

 

Now let's apply this to your examples:

 

(1)    x² + 14x +24

Start with the constant term to look for candidates for b & d. What pair of numbers multiplied together will give you a positive 24?  2 & 12, -2 & -12, 3 & 8, -3 & -8, 4 & 6, -4 & -6.

Now look at the x² coefficient, and what 2 numbers (a & c) multiplied together would produce a positive 1? Obviously: 1 times 1 or -1 times -1 are the only candidates here.

Now you look at the x term which is +14. Since the x² term's coefficient is +1, our cross term coefficient reduces to either +(b + d) or -(b + d).  So there is no way that the factors of the constant term 4 & 6 or 3 & 8 can add or subtract to produce +14. Discard them. That leaves 2 & 12 or their negatives. Since 14  & 24 both have plus signs, discard the negative versions, leaving you the factorization solution:

  (x + 12)(x + 2)

 

All this stuff can be done mentally with practice.  For your remaining problems, their solutions are:

 

2x² - 7x -15 = (2x + 3)(x - 5 ) ,  or if your problem really was 2x -7x -15, then = -5x - 15 = -5(x + 3)

x² -6x + 9   = (x - 3)(x - 3) = (x - 3)².

x² -144       = (x + 12)(x - 12)

X² - 25        = (x + 5)(x - 5)

2x² + 7x + 3 = (2x + 1)(x + 3)

 

There are polynomials that cannot be factored or simplified further because the required multiplications and additions of those a, b, c & d numbers don't work out.