In Exercises 29–40, add the polynomials. Assume that all variable exponents represent whole numbers.(5x²y + 9xy + 12) + (−3x²y + 6xy + 3)

Verified step by step guidance

1. Identify the like terms in each polynomial. Like terms have the same variables raised to the same powers.
2. Group the like terms together. For example, group the terms with \(x^2y\), the terms with \(xy\), and the constant terms.
3. Add the coefficients of the like terms. For the \(x^2y\) terms, add \(5\) and \(-3\). For the \(xy\) terms, add \(9\) and \(6\). For the constant terms, add \(12\) and \(3\).
4. Write the resulting polynomial by combining the sums of the coefficients with their respective variables.
5. Ensure that the final expression is simplified and all like terms are combined.

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables raised to whole number exponents, combined using addition, subtraction, and multiplication. Each term in a polynomial is made up of a coefficient and a variable part. For example, in the polynomial 5x²y, 5 is the coefficient, and x²y is the variable part. Understanding the structure of polynomials is essential for performing operations like addition.

Like Terms

Like terms are terms in a polynomial that have the same variable parts raised to the same powers. For instance, in the expression 5x²y and -3x²y, both terms are like terms because they share the same variable components (x²y). When adding polynomials, only like terms can be combined, which simplifies the expression and is crucial for accurate calculations.

Combining Polynomials

Combining polynomials involves adding or subtracting their respective terms. This process requires identifying and grouping like terms, then summing their coefficients. For example, when adding (5x²y + 9xy + 12) and (−3x²y + 6xy + 3), you would combine 5x²y and -3x²y, 9xy and 6xy, and the constants 12 and 3 separately. Mastery of this concept is vital for solving polynomial equations.