By Sly Tutor
Updated Aug 30, 2022
A polynomial contains only positive‑integer exponents, whereas more advanced algebraic expressions may involve fractional or negative exponents. For fractional exponents, the numerator behaves like a standard exponent and the denominator indicates the root type. Negative exponents mirror regular exponents but move the term to the denominator. Factoring such expressions requires both fraction manipulation skills and solid factoring techniques.
Identify every term that carries a negative exponent. Rewrite each as a positive exponent and transfer it to the opposite side of the fraction bar. For example, x-3 becomes 1/(x3), and 2/(x-3) turns into 2·x3. Applying this to 6(xz)2/3 – 4/[x-3/4] gives 6(xz)2/3 – 4x3/4.
Determine the greatest common divisor of all numeric coefficients. In our example, the coefficients 6 and 4 share a common factor of 2.
Divide each term by the common factor from Step 2 and place the factor outside the brackets. Factoring out 2 from the rewritten expression yields:
2[3(xz)2/3 – 2x3/4]
Locate variables that appear in every term inside the brackets. Select the term where that variable has the smallest exponent. Here, x appears in both terms, while z does not. We choose 3(xz)2/3 because 2/3 < 3/4.
Factor out the variable with the lowest exponent (excluding its coefficient). Compute the exponent difference using a common denominator:
x3/4 ÷ x2/3 = x3/4 – 2/3 = x9/12 – 8/12 = x1/12
Combine the results to write the fully factored expression:
(2)·x2/3[3z2/3 – 2x1/12]This final form illustrates the complete factorization of the original expression.
