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Chapter 5: Problem 66

Add \(15 m-9\) \(4 m+12\) \(\hline\)

Short Answer

Expert verified
The sum is Step 1: Align Terms15m - 9+ 4m + 12Step 2: Add the Coefficients15m + 4m = 19m-9 + 12 = 3Step 3: Write the ResultThe final polynomial is 19m + 3.

Step by step solution

01

Align Terms

Align the terms of the polynomials by their like terms. Place the polynomials vertically with each term in the same column.
02

Add the Coefficients

Add the coefficients of the like terms. For the terms involving 'm', add the coefficients 15 and 4. For the constant terms, add -9 and 12.
03

Write the Result

Combine the results from Step 2 to write the final polynomial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

like terms
When adding polynomials, it's important to identify and group 'like terms'. Like terms are terms within the polynomials that have the same variables raised to the same powers. In the exercise, '15m' and '4m' are like terms because they both contain the variable 'm' with the same exponent of 1. Similarly, the constants -9 and 12 are also like terms. Grouping like terms makes it easier to combine them. First, align the polynomials so that like terms are in the same column. Then, add the coefficients of the like terms together. This step helps simplify the addition of polynomials and ensures accuracy in your final result.
coefficients
Coefficients are the numerical parts of the terms in a polynomial. They multiply the variables. In the exercise, '15' is the coefficient of 'm' in the term '15m', and '4' is the coefficient of 'm' in '4m'. When adding polynomials, add the coefficients of the like terms together, while keeping the variable part unchanged. For example, sum the coefficients 15 and 4 to get 19, resulting in the term '19m'. Similarly, add the constant coefficients -9 and 12. Constants are like terms with no variables attached. The correct summing of coefficients is crucial when combining like terms, as it determines the final polynomial.
polynomials
A polynomial is an algebraic expression made up of terms. Each term consists of a variable, usually combined with a coefficient, and constants. Polynomials can have one or more terms, such as monomials (one term), binomials (two terms), and trinomials (three terms). In the given exercise, two binomials are being added. The initial polynomials are '15m-9' and '4m+12'. To add these, first, identify like terms and align them. Then, add their coefficients. Polynomials are important in mathematics because they are used in various equations and functions, making understanding their addition and manipulation foundational in algebra.

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Most popular questions from this chapter

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \frac{\left(3 r^{2}\right)^{2} r^{-5}}{r^{-2} r^{3}}\left(2 r^{-6}\right)^{2} $$

Find each product. $$ (2 a+b)\left(3 a^{2}+2 a b+b^{2}\right) $$

Write each number in scientific notation. $$ 0.875 $$

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(\frac{p^{2} q^{-1}}{2 p^{-2}}\right)^{2} \cdot\left(\frac{p^{3} \cdot 4 q^{-2}}{3 q^{-5}}\right)^{-1} \cdot\left(\frac{p q^{-5}}{q^{-2}}\right)^{3} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ \frac{(2 k)^{2} m^{-5}}{(k m)^{-3}} $$
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