UNIT 5
Polynomials and Properties of Exponents
LESSON 6
Polynomials and Properties of Exponents Review
Polynomials are algebraic expressions that involve addition, subtraction, multiplication, and non-negative integer exponents. They can be written in standard form, with the terms ordered by degree from highest to lowest.
The properties of exponents can be used to simplify and manipulate polynomial expressions. These properties include:
1. Product Rule: a^n * a^m = a^(n+m)
2. Quotient Rule: a^n / a^m = a^(n-m)
3. Power Rule: (a^n)^m = a^(n*m)
4. Negative Exponent Rule: a^(-n) = 1/a^n
5. Zero Exponent Rule: a^0 = 1
To simplify polynomial expressions, we can also use the distributive property, combine like terms, and factor out common factors. Factoring can also be used to find the roots or zeros of a polynomial, which are the values of x that make the polynomial equal to zero.
In summary, understanding polynomials and the properties of exponents is important in algebraic problem-solving and manipulation. Practice and mastery of these concepts is key to success in higher-level math courses.
In this lesson, we will review some important concepts related to polynomials and properties of exponents.
1. Polynomials:
- A polynomial is an algebraic expression with one or more terms. Each term consists of a coefficient multiplied by a variable raised to a non-negative integer power.
- The degree of a polynomial is the highest power of the variable in any of its terms.
- The standard form of a polynomial arranges the terms in order of decreasing powers of the variable.
- The leading coefficient is the coefficient of the term with the highest power.
2. Adding and Subtracting Polynomials:
- To add or subtract polynomials, we combine like terms. Like terms have the same variable raised to the same power.
- Add or subtract the coefficients of the like terms while keeping the variable and its power the same.
3. Multiplying Polynomials:
- To multiply polynomials, we use the distributive property. Multiply each term of one polynomial by each term of the other polynomial and combine like terms.
- The FOIL method is often used to multiply two binomials. Multiply the First terms, then the Outer terms, the Inner terms, and finally the Last terms.
4. Properties of Exponents:
- Exponents are used to represent repeated multiplication.
- The product of two exponential expressions with the same base can be simplified by adding their exponents.
- The quotient of two exponential expressions with the same base can be simplified by subtracting their exponents.
- When raising a power to a power, multiply the exponents.
- Any non-zero number raised to the power of 0 is equal to 1.
- Negative exponents can be rewritten as fractions with the base raised to a positive exponent.
These are the main concepts covered in this review of polynomials and properties of exponents. It is important to practice these concepts and solve various exercises to have a strong understanding.