Problem 13 - Algebra and Trigonometry

Question

In Exercises 9–14, 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 is \(6x^2-6x+2\) and its degree is 2.

Step 1: Add or Subtract similar terms

First, we add or subtract similar terms in the polynomial. We follow the rule of order of operations (PEMDAS/BODMAS) and the distributive property. So, this yields \(5x^2 + 2x^2 - x^2\) for the x^2 terms, \(-7x -3x +4x\) for the x terms, and \(-8 +7 +3\) for the constant terms.

Step 2: Compute Result

In the second step, we calculate by performing addition or subtraction. This equals to \(6x^2 -6x +2\).

Step 3: Find the degree of the polynomial

The degree of the polynomial is the highest power of x in the polynomial. Hence, the degree in this case is 2, as the highest power of x here is 2.

Key Concepts

Standard Form of Polynomial

When working with polynomials, it's important to write them in a way that makes solving and understanding them easier. This format is known as the "standard form". A polynomial is written in standard form when its terms are ordered from highest to lowest degree in terms of the variable.
For example, in the expression \( 6x^2 - 6x + 2 \), you can see that the terms are ordered from \(x^2\) to \(x\) to the constant. Each term is arranged so that the power of \(x\) decreases as you move from left to right.
Writing polynomials in standard form can help identify important aspects, like the degree, and simplifies operations like addition and subtraction.

Degree of Polynomial

The degree of a polynomial is a crucial concept that helps determine its complexity and behavior. It is simply the highest power of the variable \(x\) that appears in the polynomial. In the polynomial \(6x^2 - 6x + 2\), the highest power of \(x\) is \(x^2\), so the degree is 2.
Knowing the degree gives insights into various characteristics, such as the potential number of solutions or roots and the general shape of its graph.
Degrees dictate how a polynomial behaves as \(x\) becomes very large or very small, making them key in understanding the polynomial's overall behavior.

Order of Operations

When performing operations like adding or subtracting polynomials, following the correct order of operations is essential to avoid mistakes. Generally, this follows the PEMDAS/BODMAS rule, which means Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right).
Let's take a closer look at the expression: \((5x^2 - 7x - 8) + (2x^2 - 3x + 7) - (x^2 - 4x - 3)\).
Following the order, you first handle what's inside any parentheses, taking note of any subtraction signs that affect terms.
Next, add or subtract coefficients of like terms to simplify further. Understanding this sequence ensures that you perform operations accurately.

Distributive Property

The distributive property is a useful tool in simplifying polynomial expressions. It states that distributing a factor to each term inside parentheses will yield the same result as multiplying the factor by the entire grouped expression. This property helps when dealing with terms in addition or subtraction across different parts of an expression.
In the exercise, part of simplifying \((5x^2 - 7x - 8) + (2x^2 - 3x + 7) - (x^2 - 4x - 3)\) involves applying this property when rearranging and combining like terms.
The distributive property allows you to manage parentheses and distribute signs across terms, ensuring accurate simplification. It is a foundational concept that's widely used in combining and rearranging polynomials.