Problem 21 - College Algebra with Modeling and Visualization

Question

Add the polynomials. $$\left(2 x^{2}-x+7\right)+\left(-2 x^{2}+4 x-9\right)$$

Step-by-Step Solution

Answer

3x - 2

Step 1: Identify Like Terms

In the polynomials, identify terms with the same power of \(x\). The first polynomial is \(2x^2 - x + 7\), and the second polynomial is \(-2x^2 + 4x - 9\). The like terms are those with \(x^2\), \(x\), and constant (without \(x\)).

Step 2: Combine Like Terms with \(x^2\)

Combine the \(x^2\) terms from both polynomials. This gives us \(2x^2 - 2x^2 = 0\).

Step 3: Combine Like Terms with \(x\)

Combine the \(x\) terms: \(-x + 4x = 3x\).

Step 4: Combine Constant Terms

Combine the constant terms: \(7 - 9 = -2\).

Step 5: Write the Resulting Polynomial

Combine all the results: Since the \(x^2\) terms cancel each other, the resulting polynomial is \(3x - 2\).

Key Concepts

Like Terms

In algebra, when dealing with polynomials, recognizing "like terms" is a crucial skill. Like terms are terms that contain the same variable raised to the same power. For example, in the polynomial expressions, the terms with \(x^2\) are like terms, and so are the terms with \(x\), and the constant numbers without any variable.

Understanding like terms is essential because it allows us to simplify complex algebraic expressions by adding or subtracting them. Let's break it down further:

• Terms with \(x^2\): These terms have the variable \(x\) raised to the power of 2.
• Terms with \(x\): These are simply linear terms found in our expression.
• Constant Terms: These are standalone numbers without any accompanying variables.

Identifying like terms is the first step to simplifying polynomial expressions, as it allows you to combine them effectively.

Combining Algebraic Expressions

Once like terms are identified, combining algebraic expressions becomes straightforward. The process involves adding or subtracting coefficients of the like terms to simplify the polynomial further.

Here's a step-by-step breakdown:

• Combine Terms with \(x^2\): Start by adding or subtracting the coefficients of \(x^2\). In our example, \(2x^2\) and \(-2x^2\) cancel each other out because their sum is zero.
• Combine Terms with \(x\): Similarly, sum up the coefficients of the linear terms. In the expression, \(-x + 4x\) results in \(3x\).
• Combine Constant Terms: Lastly, add the constant numbers: \(7\) and \(-9\), which give \(-2\).

By combining the terms appropriately, you transform an initial complicated expression into a more manageable one.

Constant Terms

Constant terms in a polynomial are those that do not contain any variables. They are just numbers that stand alone in the expression.

For example, in the given polynomials, the constants are \(7\) and \(-9\). Constant terms are directly added or subtracted just like regular numbers:

• Add or Subtract Normally: Since they have no variables, treat them like simple arithmetic addition or subtraction: \(7 - 9\) results in \(-2\).

Constant terms determine the shift of the polynomial along the y-axis in a graph, affecting the intercept without influencing the slope or the nature of the curve.

Understanding the role of constant terms helps in accurately evaluating and simplifying algebraic expressions.