Problem 13 - Algebra 1: Concepts and Skills

Question

Find the sum or the difference of the polynomials. $$ \left(x^{2}-4 x+3\right)+\left(3 x^{2}-3 x-5\right) $$

Step-by-Step Solution

Answer

The sum of the given polynomials is \(4x^{2}-7x-2\).

Step 1: Identify like terms

Just observe the two polynomials. In the first polynomial, \(x^{2}\) matches with \(3x^{2}\) in the second polynomial, these are like terms. Similarly, \(-4x\) in the first polynomial matches with \(-3x\) in the second polynomial. Lastly, the constant term \(3\) in the first polynomial is a like term with \(-5\) in the second polynomial.

Step 2: Add the like terms

Now add the like terms. The sum of \(x^{2}\) and \(3x^{2}\) is \(4x^{2}\). The sum of \(-4x\) and \(-3x\) is \(-7x\). The sum of \(3\) and \(-5\) is \(-2\).

Step 3: Write the sum polynomial

The sum of the above three results is \(4x^{2}-7x-2\), which is the sum of the two original polynomials.

Key Concepts

Like Terms

When dealing with polynomials, one of the first and crucial steps is identifying like terms. Like terms are terms that have the same variables raised to the same powers. For example, terms like

• \(x^2\) and \(3x^2\)
• \(-4x\) and \(-3x\)

have the same variable parts. So, they are considered like terms. Identifying these terms is essential because it allows us to simplify polynomials effectively. In our exercise, observe how the terms with \(x^2\) in both polynomials are paired together, as well as the terms with just \(x\), and finally the constants. This organization into groups of like terms helps in methods like addition, as we'll explore next.

Adding Polynomials

Now that we've identified the like terms in polynomials, the next step is to add them together. Addition of polynomials involves combining the coefficients of like terms. Let's take an example from our exercise:

• Add \(x^2\) and \(3x^2\): The result is \(4x^2\).
• Add \(-4x\) and \(-3x\): This gives us \(-7x\).
• Finally, add \(3\) and \(-5\): You get \(-2\).

So, when adding polynomials, you essentially line up the like terms, and add their coefficients together. This results in a new polynomial where everything is neatly combined. Remember, because these are like terms, you're only adding the numbers (coefficients) in front of the variables or the constants.

Sum and Difference of Polynomials

Understanding how to calculate the sum or difference of polynomials allows you to handle more complex expressions. The sum of the polynomials from our problem forms a new polynomial: \(4x^2 - 7x - 2\). Likewise, to find the difference, you'd subtract the coefficients of the like terms instead of adding them.
Finding the difference is similar to adding, but you will subtract each term instead:

• Subtract \(3x^2\) from \(x^2\).
• Subtract \(-3x\) from \(-4x\).
• Subtract \(-5\) from \(3\).

Thus, calculating with polynomials, whether finding a sum or a difference, follows a consistent process of aligning like terms and carefully combining their coefficients. Once mastered, this becomes a powerful tool for simplifying and solving polynomial expressions.