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Polynomials are algebraic expressions that consist of variables and coefficients, combined together using addition, subtraction, and multiplication. Each term in a polynomial is made up of a coefficient (a number) and a variable raised to a power (exponent). For example, in the polynomial expression given in the exercise, you can find terms like \(13x^4\) and \(-12x^3\).
Polynomials are classified based on the number of terms they have:

• A monomial has one term (e.g., \(7x^2\)).
• A binomial has two terms (e.g., \(3x + 5\)).
• A trinomial has three terms (e.g., \(2x^2 + 3x - 4\)).

Understanding polynomials is essential in algebra as they form the basis for more complex equations and functions.

An algebraic expression is a mathematical phrase that can include numbers, variables, and operational symbols (+, -, *, /). Unlike equations, algebraic expressions don't contain an equal sign. The given exercise is an example of combining like terms in an algebraic expression.
When working with algebraic expressions, it's crucial to identify and combine like terms. Like terms are terms that have the same variable raised to the same power. For instance, \(13x^4\) and \(9x^4\) are like terms because they both involve \(x\) raised to the fourth power.
Combining like terms helps simplify the expression, making it easier to work with. This process is essential for solving equations and for manipulating algebraic expressions in advanced mathematics.

In an algebraic expression, the coefficient is the numerical part of a term with a variable. For example, in the term \(13x^4\), the coefficient is 13. Coefficients are the numbers you combine when you add or subtract like terms.
Identifying coefficients correctly is vital for accurately combining like terms. In the exercise, two terms with \(x^4\) have coefficients 13 and 9. By adding these coefficients, you get \(22x^4\). Similarly, the terms with \(x^3\) have coefficients -12 and 2, which combine to give \(-10x^3\).
Simplifying algebraic expressions by combining like terms reduces the number of terms and streamlines calculations, making it easier to solve equations and understand complex problems.