A polynomial is an algebraic expression consisting of variables and coefficients, where the variables are only raised to non-negative integer powers. Polynomials are fundamental building blocks in algebra and are central to many topics in elementary algebra.
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Polynomials can be added and subtracted by combining like terms, as covered in topic 6.1.
Polynomials can be multiplied using the distributive property, as discussed in topic 6.3.
Special products of polynomials, such as the difference of two squares and the square of a binomial, are covered in topic 6.4.
Factoring polynomials, including finding the greatest common factor and factoring trinomials, is the focus of topics 7.1 through 7.5.
Quadratic equations, a specific type of polynomial equation, are explored in topic 7.6 and their graphs are discussed in topic 10.5.
Review Questions
Explain how the concept of a polynomial relates to the process of adding and subtracting polynomials.
The defining feature of a polynomial is that it is an algebraic expression with variables raised to non-negative integer powers and coefficients. When adding or subtracting polynomials, as covered in topic 6.1, the key is to identify and combine like terms, which are the individual parts of the polynomial with the same variable and exponent. Understanding the structure of a polynomial is essential for performing these basic algebraic operations effectively.
Describe how the properties of polynomials are utilized in the process of multiplying polynomials, as discussed in topic 6.3.
The multiplication of polynomials, as covered in topic 6.3, relies on the distributive property of multiplication. This property states that the product of a sum is equal to the sum of the individual products. By applying this principle, students can multiply polynomials by distributing each term of one polynomial to the other, resulting in a new polynomial that combines the terms in a systematic manner. Mastering the multiplication of polynomials is a crucial skill for working with more complex algebraic expressions.
Analyze how the concept of a polynomial is central to the process of factoring polynomials, as explored in topics 7.1 through 7.5.
Factoring polynomials, which is the focus of topics 7.1 through 7.5, involves breaking down a polynomial expression into a product of simpler polynomial expressions. This process relies on the fundamental structure of a polynomial, as it requires identifying the greatest common factor, recognizing the form of the trinomial, and applying strategies to isolate the individual terms. Understanding the properties and characteristics of polynomials is essential for successfully factoring these algebraic expressions, which is a key skill for solving more advanced equations and problems.
A binomial is a polynomial with two terms, typically in the form $ax^n + bx^m$, where $a$ and $b$ are coefficients, and $n$ and $m$ are non-negative integers.