Product Rule of Exponents
Multiplication of the same constant or variable to various powers can be simplified by adding up their exponents:
This also holds true for distribution:
Power Rule of Exponents
Exponents distribute with multiplication:
Signs are to be interpreted as :
The product rule is not used when constants are not the same.
Polynomials and Monomials
Polynomials are expressions linking multiple terms by addition and/or subtraction.
The degree of a term is the sum of all of its exponents. A polynomial's degree is the highest degree of the degrees of its terms.
An expression containing any of the following is not a polynomial:
- a fraction with a variable in the denominator
- any negative exponents
Distribution of Monomials
Distribution of Binomials
All terms are to be multiplied together with the resulting terms linked by either addition or subtraction:
Special Products of Polynomials
Dividing Polynomials by Monomials
Considering that , divisions of polynomials by monomials can be interpreted as the sum of two fractions with the same denominator:
Long Division Algorithm for Polynomials
In some cases long division is the best way to handle the division of a polynomial by another polynomial:
- Fill up missing powers
- Divide the first term on the right side () by the first term on the left side ()
- Move result to the top of the line
- Distribute the same term with the terms on the left side, writing the result below the term on the right (while filling up missing powers)
- Subtract the polynomial below the line with the new one written beneath it
- Bring down the next term if it isn't already part of the result
- If the algorithm's last subtraction does not yield a , it has yielded a remainder. The remainder is brought up as numerator over the left polynomial (in this case as its denominator.
Negative exponents can be transformed into positive numbers by turning their base into the numerator of a fraction and turning that fraction into its reciprocal:
Quotient Rule for Exponents
A number raised to a power divided by the same number is equal to that itself raised to the power of the exponents' difference.
Complex terms with fractions
These can be interpreted as multiple individual fractions multiplied together: