91Ó°ÊÓ

Polynomials are algebraic expressions that consist of variables and coefficients. They involve operations of addition, subtraction, multiplication, and positive integer exponents of variables. Each term in a polynomial is composed of a constant multiplied by a variable raised to a power. For example, in the polynomial \(4b^3 + b^2 + 5b + 7\), the terms are 4b^3, b^2, 5b, and 7. Polynomials can have one or more terms, and the degree of a polynomial is determined by the highest power of the variable in the expression. Example terms in a polynomial include:
\(3x^2\) with a degree of 2
\(-5x\) with a degree of 1
\(6\), which is a constant term
Combining like terms within a polynomial requires finding terms with the same variable raised to the same power and then performing operations to simplify the expression.

Adding algebraic expressions, including polynomials, involves combining like terms to simplify the expressions. In the given exercise, we add \( b^2 + 5b + 13 \) and \( 4b^3 - 6 \). These steps involve:
Identifying each term in both expressions
Writing the expressions in a way that aligns similar terms for easy addition
In this case, there were no like terms between the two expressions, so we simply add them:
\((4b^3 - 6) + (b^2 + 5b + 13) = 4b^3 + b^2 + 5b + 13 - 6 = 4b^3 + b^2 + 5b + 7\)
Keep in mind that understanding polynomial structure helps in identifying like terms and performing addition correctly.

Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. For instance, in polynomials, \(2x\) and \(5x\) are like terms, whereas \(2x\) and \(3x^2\) are not. The steps to combine like terms involve:
Grouping all like terms together
Adding or subtracting the coefficients of the like terms
In our exercise, the expressions provided do not have like terms. We have an expression \( b^2 + 5b + 13 \) and another \( 4b^3 - 6 \). Since they do not share any terms with the same power, we simply combine them as they are:
\(4b^3 + b^2 + 5b + 7\)
This results in a single polynomial expression that encompasses all the terms.