UNIT 5
Polynomials and Properties of Exponents
LESSON 1
Polynomials
A polynomial is a mathematical expression consisting of variables and coefficients, and is formed by adding or subtracting terms.
For example,
3x^2 + 2x - 5
is a polynomial consisting of three terms. The variable is x, the coefficients are 3, 2, and -5, and the exponents are 2, 1 (which is usually not written explicitly), and 0 respectively.
The degree of a polynomial is the highest exponent present in the expression. In the example above, the degree is 2.
Polynomials can be added or subtracted by combining like terms. For example,
(3x^2 + 2x - 5) + (2x^2 + 3x + 1)
can be simplified by adding the like terms:
= (3x^2 + 2x^2) + (2x + 3x) + (-5 + 1)
= 5x^2 + 5x - 4
Polynomials can also be multiplied, using the distributive property. For example,
(3x^2 + 2x - 5)(2x + 1)
= 6x^3 + 3x^2 + 4x^2 + 2x - 10x - 5
= 6x^3 + 7x^2 - 8x - 5
Polynomials are important in many areas of mathematics, including algebra, calculus, and number theory. They are used to model and solve a wide variety of problems, and have many important properties and applications.
In Lesson 1 of Unit 5, we will be learning about polynomials. A polynomial is a mathematical expression that consists of variables, coefficients, and exponents. Let's take a closer look at the components of a polynomial:
1. Variables: These are symbols or letters usually represented by x, y, or z, which represent unknown quantities in the polynomial.
2. Coefficients: These are the numbers that multiply the variable(s) in the polynomial. For example, in the term 3x, the coefficient is 3.
3. Exponents: These are the powers to which the variable(s) are raised. For example, in the term 3x^2, the exponent is 2.
Polynomials can have one or more terms. A term is a single part of a polynomial separated by addition or subtraction symbols. For example, in the polynomial 3x^2 + 2xy - 5, there are three terms: 3x^2, 2xy, and -5.
Polynomials can be classified by their degree, which is the highest exponent of the variable in the polynomial. The degree determines the shape and behavior of the polynomial.
Some common polynomial degrees include:
1. Constant: A polynomial with a degree of 0, such as 5 or -2. These polynomials have no variable terms and are essentially just numbers.
2. Linear: A polynomial with a degree of 1, such as 3x + 2 or -4y + 7. These polynomials have only one variable term.
3. Quadratic: A polynomial with a degree of 2, such as 4x^2 + 5x - 2. These polynomials have a variable term raised to the power of 2.
4. Cubic: A polynomial with a degree of 3, such as 2x^3 - 3x^2 + x + 1. These polynomials have a variable term raised to the power of 3.
In Lesson 1, we will learn how to identify the degree of a polynomial, simplify polynomials by combining like terms, and perform operations such as addition, subtraction, and multiplication on polynomials.
Polynomials are mathematical expressions made up of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication operations. They can be written in the form:
P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₂x² + a₁x + a₀
Here, P(x) represents the polynomial function, n is a non-negative integer, aₙ, aₙ₋₁, ..., a₂, a₁, a₀ are the coefficients, and x is the variable.
To identify a polynomial, there are a few key properties to look for:
1. The exponents on the variable are non-negative integers.
2. The coefficients can be any real numbers.
3. The terms are added or subtracted together, but they are not multiplied or divided.
To determine the degree of a polynomial, you need to find the term with the highest exponent. The degree of the polynomial is equal to the value of that exponent. For example, if the highest exponent is 3, the polynomial is a cubic polynomial, and if the highest exponent is 2, it is a quadratic polynomial.
To add or subtract polynomials, you combine like terms. Like terms have the same variable raised to the same exponent. For example, 3x² and -2x² are like terms, but 3x² and 3x are not.
When multiplying polynomials, you use the Distributive Property. You multiply each term in one polynomial by each term in the other polynomial and then combine like terms.
Understanding polynomials and their properties is essential in algebra and calculus, as they are used to solve equations, graph functions, and perform other mathematical operations.