Higher Degrees and Variable Exponents
It is not necessary always to use substitution to factor polynomials with higher
degrees or variable exponents. In the next example we use
trial and error to factor two polynomials of higher degree and one with variable
exponents. Remember that if there is a common factor to all terms, factor it out first.
Example 1
Higherdegree and variable exponent trinomials
Factor each polynomial completely. Variables used as exponents represent positive
integers.
a) x^{8}  2x^{4}  15
b) 18y^{7} + 21y^{4} + 15y
c) 2u^{2m}  5u^{m}  3
Solution
a) To factor by trial and error, notice that x^{8} = x^{4} Â·
x^{4}. Now 15 is 3 Â· 5 or
1 Â· 15. Using 1 and 15 will not give the required 2 for the coefficient of the
middle term. So choose 3 and 5 to get the 2 in the middle term:
x^{8}  2x^{4}  15 = (x^{4}  5)(x^{4} + 3)
b) 18y^{7} + 21y^{4} + 15y 
= 3y(6y^{6}  7y^{3}  5) 
Factor out the common factor 3y first. 

= 3y(2y^{3} + 1)(3y^{3}  5) 
Factor the trinomial by trial and error. 
c) Notice that 2u^{2m} = 2u^{m} Â· u^{m} and 3 = 3 Â· 1. Using trial and error, we get
2u^{2m}  5u^{m}  3 = (2u^{m} + 1)(u^{m}
 3).
