Problem 13 - College Algebra

Question

Perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$\left(5 x^{2}-7 x-8\right)+\left(2 x^{2}-3 x+7\right)-\left(x^{2}-4 x-3\right)$$

Step-by-Step Solution

Answer

The resulting polynomial in standard form is \(6x^2 - 14x - 4\) and its degree is 2.

Step 1: Combine Similar Terms

Firstly, distribute the negative sign into the third polynomial, which changes the equation to \((5 x^{2}-7 x-8) + (2 x^{2}-3 x +7) - (x^{2}-4 x-3)\). Next, combine the terms of each polynomial that contain the same variable raised to the same power. Thus, combine the \(x^2\) terms, the \(x\) terms, and the terms with no variable. The addition of the \(x^2\) terms will form \(6x^2\), the addition of \(x\) terms will form \(-14x\), and the addition of the constant terms will form \(-4\). Thus the equation will simplify to \(6x^2 - 14x - 4\).

Step 2: Write in Standard Form

After combining similar terms, the resulting polynomial can now be written in standard form. In standard form, terms are ordered from highest degree to lowest degree. The resulting expression, \(6x^2 - 14x - 4\), is already in this form.

Step 3: Identify the Degree

The degree of a polynomial is the highest power of the variable that appears in the polynomial. In this case, the highest power is 2, so the degree of the polynomial is 2.

Key Concepts

Combining Like Terms

When working with polynomials, you often need to combine like terms to simplify expressions. This involves grouping terms with the same variable and exponent. For example, in the polynomial expression \((5x^2 - 7x - 8) + (2x^2 - 3x + 7) - (x^2 - 4x - 3)\), you start by distributing any negative signs to accurately adjust each term.
After that, you should:

• Combine all the \(x^2\) terms: These are \(5x^2\), \(2x^2\), and \(-x^2\). Adding these gives \(6x^2\).
• Combine all the \(x\) terms: These are \(-7x\), \(-3x\), and \(-4x\). Adding these equals \(-14x\).
• Combine the constant terms: These are \(-8\), \(+7\), and \(+3\). Adding these results in \(-4\).

After combining these like terms, the expression simplifies to \(6x^2 - 14x - 4\). Combining like terms helps make a polynomial easier to work with and paves the way for writing it in standard form.

Standard Form of a Polynomial

The standard form of a polynomial organizes the terms in order of decreasing degree. This is an essential aspect because it makes the polynomial easier to understand and analyze, particularly when identifying its degree.
For a polynomial to be in standard form:

• The term with the highest degree comes first.
• Subsequent terms follow in order of decreasing degree.
• Typically, the exponents of the variable decrease from left to right.

In our simplified expression \(6x^2 - 14x - 4\), the terms are already organized by degree from highest to lowest:- The term \(6x^2\) has the highest degree, so it comes first.- Next is the term \(-14x\), which has a degree of 1.- The constant term \(-4\) has a degree of 0.This arrangement ensures the polynomial is presented neatly, following the convention of standard form.

Degree of a Polynomial

The degree of a polynomial is a key characteristic that indicates the highest power of the variable in the polynomial. Understanding the degree helps in determining the behavior and potential graph of the polynomial function.
To identify the degree:

• Look for the term with the highest exponent.
• The exponent of this term is the degree of the polynomial.
• In multi-variable polynomials, sum of exponents in each term is considered.

For our expression \(6x^2 - 14x - 4\), the degree is determined by the term \(6x^2\), which has an exponent of 2. This means the degree of the polynomial is 2.
The degree tells us that this polynomial is quadratic, and it helps predict its general shape when graphed. Understanding the degree is crucial for solving polynomial equations and predicting their behavior.