Problem 29 - Elementary and Intermediate Algebra

Question

Add the polynomials. $$ \left(3 q^{2}-5 q+7\right)+\left(2 q^{2}+q-12\right) $$

Step-by-Step Solution

Answer

The sum of the polynomials is \(5q^2 - 4q - 5\).

Step 1: Identify Like Terms

First, identify the corresponding terms in each polynomial that have the same degree. In this case, the terms are: \(3q^2\) and \(2q^2\) for the quadratic terms, \(-5q\) and \(q\) for the linear terms, and \(7\) and \(-12\) for the constant terms.

Step 2: Add Quadratic Terms

Add the quadratic terms from each polynomial: \(3q^2 + 2q^2\). This equals \(5q^2\).

Step 3: Add Linear Terms

Add the linear terms from each polynomial: \(-5q + q\). This equals \(-4q\).

Step 4: Add Constant Terms

Add the constant terms from each polynomial: \(7 - 12\). This equals \(-5\).

Step 5: Write the Result

Combine the sums of each type of term to form the resulting polynomial: \(5q^2 - 4q - 5\).

Key Concepts

Like Terms

When working with polynomials, identifying and understanding like terms is crucial. Like terms are terms that have the same variable raised to the same power. In our exercise, terms are grouped based on their degree or the exponent of their variable. This makes it easier to combine them. For example:

• Quadratic terms have the variable raised to the power of 2, such as \(3q^2\) and \(2q^2\).
• Linear terms have the variable raised to the power of 1, like \(-5q\) and \(q\).
• Constant terms are numbers without any variables, such as \(7\) and \(-12\).

Once these like terms are identified, they can be easily added or subtracted to simplify the polynomial expression.

Quadratic Terms

Quadratic terms are terms where the variable is squared (i.e., raised to the power of 2). In polynomial expressions, these terms are combined first by adding or subtracting them, as they typically have a greater impact on the shape of the graph if we were to visualize it.
For our exercise, the quadratic terms are \(3q^2\) and \(2q^2\). To combine these, we simply add their coefficients:

• Calculation: \(3q^2 + 2q^2 = 5q^2\)

This result becomes part of the larger polynomial solution. It's important to remember that only the coefficients are added, while the variable and its exponent stay the same.

Linear Terms

Linear terms in a polynomial involve a variable raised to the first power. They appear as terms like \(-5q\) and \(q\) in our example. Linear terms directly affect the slope of the line if we were to graph the polynomial.
Combining linear terms follows the same process of adding or subtracting their coefficients:

• Calculation: \(-5q + q = -4q\)

Here, the coefficients \(-5\) and \(1\) (an implicit one in front of \(q\)) are combined, resulting in \(-4q\). This term reflects the change in slope when altering the overall polynomial.

Constant Terms

Constant terms are simple numbers that do not accompany any variables. In the context of polynomials, they can be thought of as a vertical shift in a graph. In our exercise, they appear as \(7\) and \(-12\).
Adding or subtracting constants is straightforward, as we only deal with numbers:

• Calculation: \(7 - 12 = -5\)

The resulting constant, \(-5\), contributes to the final polynomial. Constant terms provide the baseline or starting point on the y-axis if graphed, making them a pivotal part of understanding the entire expression.