Polynomial Expressions

Definition

A polynomial expression is a mathematical expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers and are combined using the operations of addition, subtraction, and multiplication. Polynomial expressions are fundamental in the context of adding and subtracting rational expressions, as they form the building blocks of these operations.

5 Must Know Facts For Your Next Test

1. Polynomial expressions can be added and subtracted by combining like terms, which are terms with the same variable raised to the same power.
2. The degree of a polynomial expression determines the complexity of the expression and the type of operations that can be performed on it.
3. Polynomial expressions can be factored to simplify them and make operations like addition and subtraction more efficient.
4. Rational expressions, which are the focus of the 7.2 Add and Subtract Rational Expressions topic, are formed by dividing one polynomial expression by another.
5. Understanding the properties and operations of polynomial expressions is crucial for mastering the skills required to add and subtract rational expressions.

Review Questions

• Explain how the degree of a polynomial expression relates to the complexity of the expression and the operations that can be performed on it.
  • The degree of a polynomial expression is the highest exponent of the variable in the expression. This degree determines the complexity of the polynomial and the types of operations that can be performed on it. Polynomials with lower degrees are generally simpler and easier to work with, while those with higher degrees are more complex. For example, a linear polynomial (degree 1) can be easily graphed and solved, while a quadratic polynomial (degree 2) requires more advanced techniques like factoring or the quadratic formula. Understanding the degree of a polynomial is crucial for selecting the appropriate methods and strategies when performing operations like addition, subtraction, and rational expression manipulation.
• Describe the process of combining like terms in a polynomial expression and explain how this simplifies the expression.
  • Combining like terms in a polynomial expression involves adding or subtracting the coefficients of terms that have the same variable raised to the same power. This process simplifies the expression by reducing the number of terms and making the expression easier to work with. For example, the polynomial expression $3x^2 - 5x^2 + 2x - x$ can be combined by adding the coefficients of the $x^2$ terms ($3 - 5 = -2$) and the coefficients of the $x$ terms ($2 - 1 = 1$), resulting in the simplified expression $-2x^2 + x$. Combining like terms is an essential skill for performing operations like addition and subtraction on polynomial expressions, as it allows you to work with a more manageable and organized expression.
• Explain how the understanding of polynomial expressions is crucial for mastering the skills required to add and subtract rational expressions, as covered in the 7.2 Add and Subtract Rational Expressions topic.
  • Rational expressions, which are the focus of the 7.2 Add and Subtract Rational Expressions topic, are formed by dividing one polynomial expression by another. To effectively add and subtract rational expressions, you must have a strong understanding of the properties and operations of polynomial expressions. This includes knowing how to combine like terms, factor polynomials, and manipulate expressions with variables raised to different powers. Without a solid foundation in polynomial expressions, you will struggle to simplify rational expressions, find common denominators, and perform the necessary steps to add and subtract them. The skills and knowledge gained from studying polynomial expressions are directly applicable to the 7.2 Add and Subtract Rational Expressions topic, and mastering this key term is essential for success in that area of study.

Related terms

Monomial: A monomial is a polynomial expression with only one term, consisting of a coefficient and a variable raised to a non-negative integer power.

Binomial: A binomial is a polynomial expression with two terms, consisting of two monomials connected by the operations of addition or subtraction.

Degree of a Polynomial: The degree of a polynomial expression is the highest exponent of the variable in the expression.