could someone help me understand this?

Question

Reduce the rational expression to lowest terms. If it is already in lowest terms, enter the expression in the answer box.

(18x³-21x²-60x)/(2x²+5x-25)

 

Answer 1

 

(18x³-21x²-60x)/(2x²+5x-25)

 

NUMERATOR:

All terms are factorable by 3x so numerator is:

3x(6x²-7x-20)

 

The polynomial (6x²-7x-20) is factorable by grouping.

Multiply the coefficient of the square term, 6, by the

constant, -20.   6(-20) = -120

Look for factors of -120 that add to -7, the coefficient

of the x term.   (-15)(8)=-120 and -15+8=-7

Substitute -15x+8x in place of -7x in the polynomial.

6x²-7x-20 = 6x² -15x + 8x -20

               = 3x(2x-5) + 4(2x-5)

               = (3x+4)(2x-5)

 

Your numerator factors out to:  3x(3x+4)(2x-5)

 

DENOMINATOR:

2x²+5x-25

This is also factorable by grouping.

2(-25) = -50.  Factors of -50 that add to 5 are (10)(-5)

 

2x²+5x - 25 = 2x² +10x -5x -25

                  = 2x(x+5) -5(x+5)

                  = (2x-5)(x+5)

 

Putting this back together your polynomial is:

 

[3x(3x+4)(2x-5)]/[(2x-5)(x+5)]

 

Cancelling out the common factor of 2x -5 we get

 

[3x(3x+4)]/(x+5)