General Educational Development (GED): Exponents and Polynomials

Exponents & Polynomials

To add polynomials, we first simplify the polynomials by removing all brackets. Then, we combine like terms. Like terms are terms that share the same base and power for each variable. When you have identified the like terms, we then apply the required operation which is addition to the coefficients.

To subtract polynomials, we first simplify the polynomials by removing all brackets. This is done by distributing the negative sign to all the numbers in the second bracket. This changes the problem to an addition problem. Then, we combine like terms. 

Multiplying Polynomials

Step 1: Distribute each term of the first polynomial to every term of the second polynomial. 
Step 2: Combine like terms

Example:

Multiply: 5x²y(7x² – 4xy² + 2y³)

When dividing polynomials, we can use either long division or synthetic division to arrive at an answer. Using long division, dividing polynomials is easy. We simply write the fraction in long division form by putting the divisor outside of the bracket and the divided inside the bracket. After the polynomial division is set up, we follow the same process as long division with numbers.

 An exponent refers to the number of times a number is multiplied by itself.

• Product Rule: Whenever you multiply two terms with the same base, you can add the exponents but keep the base. a^m ∙ aⁿ = a^{m + n}
• one raised to any power is one
• Power rule: tells us that to raise a power to a power, just multiply the exponents. (a^m)ⁿ = a^mn
• The zero rule: states that any nonzero number raised to the power of zero equals 1. a⁰ = 1
• The quotient rule: tells us that we can divide two powers with the same base by subtracting the exponents.  
• Negative Exponents: Any nonzero number raised to a negative power equals its reciprocal raised to the opposite positive power.  

When adding or subtracting exponents you must first determine if the base or variable and the exponents are the same for the two terms. If they are the same then you just add or subtract the coefficients together and keep the base and exponents the same.