Problem 25 - College Algebra with Modeling and Visualization

Question

Add the polynomials. $$\left(x^{4}-3 x^{2}-4\right)+\left(-8 x^{4}+x^{2}-\frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The sum is \(-7x^{4} - 2x^{2} - \frac{9}{2}\).

Step 1: Write Down the Polynomials

We are given two polynomials: \(x^{4} - 3x^{2} - 4\) and \(-8x^{4} + x^{2} - \frac{1}{2}\). Our task is to add these polynomials.

Step 2: Align the Like Terms

Align the polynomials vertically by their like terms to make it easier to add. Like terms are terms that have the same variables raised to the same power. Arrange them like this:\[\begin{align*} x^{4} & - 3x^{2} & - 4 \-8x^{4} & + x^{2} & - \frac{1}{2}\end{align*}\]

Step 3: Add the Like Terms

Add the coefficients of like terms. Starting with the \(x^4\) terms, we have:\[x^{4} - 8x^{4} = -7x^{4}\]For the \(x^2\) terms, we have:\[-3x^{2} + x^{2} = -2x^{2}\]Finally, add the constant terms:\[-4 - \frac{1}{2} = -\frac{8}{2} - \frac{1}{2} = -\frac{9}{2}\]

Step 4: Write the Final Polynomial

Combine all the results from the previous step to write the final polynomial. The sum of the polynomials is:\[-7x^{4} - 2x^{2} - \frac{9}{2}\]

Key Concepts

Like Terms

In polynomial addition, the concept of "like terms" is crucial for simplifying expressions effectively. Like terms are terms that have the exact same variables raised to the same power. For example, in the polynomial expressions \(x^4 - 3x^2 - 4\) and \(-8x^4 + x^2 - \frac{1}{2}\), the terms \(x^4\) and \(-8x^4\) are considered like terms because they both contain the variable \(x\) raised to the power of 4.

When adding polynomials, only like terms can be combined together. You should match them by their variable parts and then add or subtract their coefficients, which we will discuss in a later section. This step is important as it simplifies the expression by reducing the number of terms. Make sure to keep the variable and its exponent unchanged while combining, only adjust the numerical part, or coefficient.

Without combining like terms, a polynomial can become complicated and difficult to manage; thus, it is an essential technique to ensure simplicity and accuracy when working with algebraic expressions.

Coefficient

In polynomials, coefficients are the numerical parts of terms. They are attached to the variables and dictate the value of each term within the expression. For instance, in the term \(-3x^2\), the coefficient is \(-3\). It's the part that multiplies the variable \(x^2\).

When you're adding polynomials, you focus on these coefficients of like terms, since these are the values you will directly combine. Consider the polynomial terms \(x^4 - 8x^4\). Here, you notice both terms share the same variable parts. Thus, you add the coefficients \(+1\) (implicitly in front of \(x^4\)) and \(-8\), resulting in \(-7x^4\).

Similarly, for \(-3x^2 + x^2\), you add \(-3\) and \(+1\) to get \(-2x^2\). Adjusting the coefficients correctly determines the resulting polynomial's form. Skipping this correctly can lead to errors in polynomial calculations and outcomes. Coefficients determine how 'heavy' or 'influential' a term is within a polynomial.

Polynomial Expression

Polynomial expressions are algebraic expressions composed of variables, coefficients, and exponents arranged in a sum of terms. Each term in a polynomial is distinguished by its variable parts, typically expressed in descending order of power, like \(x^4 - 3x^2 - 4\).

In polynomial addition, each term maintains its own characteristics despite being combined with others. This particularly includes the variables and their exponents, commonly referred to as its 'like terms', which stay intact, while the coefficients might change.

For example, the exercise involved handling the expression \((x^4 - 3x^2 - 4) + (-8x^4 + x^2 - \frac{1}{2})\). This task is fundamentally about reorganizing each part—ensuring like terms line up, to straightforwardly add their coefficients. The structure of polynomials makes it clear which parts are compatible for combination, demonstrating both the beauty and efficiency of polynomial algebra.

Understanding a polynomial expression's structure is essential because it allows one to operate confidently within algebraic manipulation, ensuring clear and accurate solutions.