Problem 43 - Beginning and Intermediate Algebra

Question

Add $$\begin{array}{r} -2.6 q^{3}-q^{2}+6.9 q-1 \\ +\quad 4.1 q^{3} \quad-2.3 q+16 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The addition of the given polynomials is \(1.5q^3 - q^2 + 4.6q + 15\).

Step 1: Identify like terms

The like terms in both polynomials are the ones with the same variable and the same exponent. So, we can rewrite both polynomials, arranging them with like terms one below the other: -2.6q³ - q² + 6.9q - 1 + 4.1q³ - 2.3q +16 ----------------------

Step 2: Add the coefficients of like terms

Now, we just need to sum up the coefficients of like terms: (-2.6q³ + 4.1q³) + (-q²) + (6.9q - 2.3q) + (-1 + 16)

Step 3: Simplify the expression

Perform the arithmetic operations to get the simplified expression: 1.5q³ - q² + 4.6q + 15 So, the addition of the given polynomials is \(1.5q^3 - q^2 + 4.6q + 15\).

Key Concepts

Like Terms

When dealing with polynomials, it's essential to identify like terms. These are terms that have the same variable raised to the same power.
For example, in the expression \(-2.6q^3 - q^2 + 6.9q - 1\) and \(4.1q^3 - 2.3q + 16\), the like terms are:

• \(-2.6q^3\) and \(4.1q^3\) (both have \(q^3\))
• \(6.9q\) and \(-2.3q\) (both have \(q^1\))
• \(-1\) and \(16\) (constant terms with no variables)

Recognizing like terms allows you to group them together, which is crucial for efficiently adding or subtracting polynomials.

Coefficients

Coefficients are the numerical part of the terms in an algebraic expression. For example, in the term \(-2.6q^3\), \(-2.6\) is the coefficient. Coefficients dictate the magnitude and direction (positive or negative) of the term.
Understanding coefficients enables us to efficiently perform operations like addition.
For instance, when we add the like terms \(-2.6q^3 + 4.1q^3\), we combine the coefficients \(-2.6\) and \(4.1\), resulting in \(1.5q^3\).
This simplification is essential to make the expressions more manageable.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and subtraction). They form the building blocks of algebra.
A polynomial is a type of algebraic expression, involving one or more terms. Each term includes a coefficient and a variable raised to a power.
For example, the expression \(1.5q^3 - q^2 + 4.6q + 15\) is a polynomial with four terms.
Understanding algebraic expressions helps students grasp complex mathematical concepts by breaking them down into simpler parts.

Simplifying Polynomials

Simplifying polynomials involves combining like terms to reduce the expression to its simplest form. This process makes it easier to work with equations and perform further operations.
In the example \(-2.6q^3 - q^2 + 6.9q - 1 + 4.1q^3 - 2.3q + 16\), we simplify by adding the like terms:

• \(1.5q^3\) from \(-2.6q^3 + 4.1q^3\)
• \(-q^2\) remains as there's no matching term
• \(4.6q\) from \(6.9q - 2.3q\)
• \(15\) from \(-1 + 16\)

This results in \(1.5q^3 - q^2 + 4.6q + 15\). Simplifying polynomials is a key skill in algebra, making further calculations more straightforward.