Problem 11 - Algebra and Trigonometry

Question

In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$ \left(17 x^{3}-5 x^{2}+4 x-3\right)-\left(5 x^{3}-9 x^{2}-8 x+11\right) $$

Step-by-Step Solution

Answer

The resulting polynomial is \(12x^{3} + 4x^{2} + 12x - 14\) and its degree is 3.

Step 1: Simplify the Expression

Given the expression \(17x^{3}-5x^{2}+4x-3\) - \(5x^{3}-9x^{2}-8x+11\). First, distribute the minus sign across the second parentheses to change the sign of each term: \(17x^{3}-5x^{2}+4x-3 - 5x^{3} + 9x^{2} + 8x - 11\).

Step 2: Combine Like Terms

Next, group and add/subtract like terms together. This results in: \(17x^{3} - 5x^{3} + 9x^{2} - 5x^{2} + 4x + 8x - 3 - 11\). If done correctly, this simplifies to: \(12x^{3} + 4x^{2} + 12x - 14\).

Step 3: Identify the Degree

Finally, the degree is the highest power of x in the polynomial. Looking at the output polynomial, the degree clearly in this case is 3, because the leading term is \(12x^{3}\).

Key Concepts

Understanding Standard Form Polynomials

When working with polynomials, it's essential to express them in standard form. In standard form, a polynomial is written as a sum of terms ordered by descending powers of the variable. Each term consists of a coefficient and a power of the variable, such as\[ax^n + bx^{n-1} + cx^{n-2} + \ldots + k,\] where \(a, b, c, \ldots , k\) are coefficients and \(n\) is a non-negative integer. This order makes it easier to identify important features of the polynomial, such as its degree. The highest exponent of the variable indicates the leading term.
In our given exercise, after simplifying, we write the polynomial as \(12x^3 + 4x^2 + 12x - 14\). Each term of the polynomial is in descending order based on the exponent of \(x\). By arranging in such a manner, not only do we achieve standard form but we also facilitate straightforward interpretation and further operations.

Determining the Degree of a Polynomial

The degree of a polynomial is one of its most potent descriptive tools. It's defined as the highest exponent of the variable in the polynomial expression when it's in standard form. The degree offers insights into the behavior of the polynomial function, such as its growth rate, number of roots, and the shape of its graph.

To determine the degree, identify the term with the largest exponent. This is the leading term. For example, in our polynomial \(12x^3 + 4x^2 + 12x - 14\), the leading term is \(12x^3\) and, thus, the degree is 3.

• A degree of 3 implies the graph of the polynomial will have an overall cubic shape.
• It suggests that the function can have up to 3 roots, though not necessarily distinct or real.

Understanding the degree is crucial for graphing and solving polynomial equations.

The Process of Combining Like Terms

In polynomial operations, combining like terms is a fundamental skill. Like terms are terms that have the same variable raised to the same power. It's important to combine these because it simplifies the expression and makes it easier to work with.
For the given expression \(17x^3-5x^2+4x-3 - (5x^3-9x^2-8x+11)\), after distributing the negative sign, we obtain, \(17x^3 - 5x^3 + 9x^2 - 5x^2 + 4x + 8x - 3 - 11\). Here's how we combine the like terms:

• Combine \(17x^3\) and \(-5x^3\) to become \(12x^3\).
• Combine \(-5x^2\) and \(9x^2\) to get \(4x^2\).
• Add \(4x\) and \(8x\) to yield \(12x\).
• The constant terms \(-3\) and \(-11\) combine to \(-14\).

This simplification process is essential to reducing polynomial expressions to a more usable form, and is critical for solving polynomial equations efficiently.