Problem 2 - Intermediate Algebra

Question

Fill in the blanks. The powers of \(x\) in \(2 x^{4}+3 x^{3}+4 x^{2}-7 x-8\) are written in _________ order.

Step-by-Step Solution

Answer

Decreasing

Step 1: Identify the powers of x

The expression given is \(2x^4 + 3x^3 + 4x^2 - 7x - 8\). Identify the powers of \(x\): \(x^4\), \(x^3\), \(x^2\), \(x^1\), and \(x^0\).

Step 2: Determine the order of powers

Look at the sequence of powers: 4, 3, 2, 1, and 0. They decrease in order from left to right.

Step 3: Conclusion on order type

Since the powers decrease from 4 to 0, they are written in decreasing order.

Key Concepts

Decreasing Order

Polynomials are often expressed in a sequence where the terms are arranged according to their degree, which is the highest power of the variable in that term. A polynomial is said to be in decreasing order when the terms are arranged from the highest power to the lowest power. In our example, the polynomial is

• 2x⁴ + 3x³ + 4x² - 7x - 8.

In this expression, the term with the highest power, which is 4, is written first, followed by terms with powers 3, 2, 1, and finally 0. It's important because it helps in systematically comparing different polynomials and performing operations like addition or subtraction. When a polynomial is given in decreasing order, it becomes easier to understand and solve related problems efficiently.

Powers of x

In the polynomial expression, every term is composed of a constant coefficient and the variable raised to a certain power, also known as the exponent. In mathematical notation, these exponents are referred to as the powers of the variable, in this case, "x". Let's look at our example expression

• 2x⁴ + 3x³ + 4x² - 7x - 8

The powers here are 4, 3, 2, 1, and 0, listed in that exact order. Here's a breakdown of each term:

• 2x⁴: Power of x is 4.
• 3x³: Power of x is 3.
• 4x²: Power of x is 2.
• -7x: Power of x is 1, as it is the same as x¹.
• -8: Power of x is 0 because it can be written as -8x⁰; any non-zero number to the zero power is 1.

Understanding these powers is crucial when performing polynomial operations such as addition, subtraction, and multiplication, as they determine each term's behavior within the expression.

Step-by-Step Solution

Solving problems involving polynomials often involves following a structured pattern or methodology. Let's explore the given exercise with a step-by-step approach:

• **Step 1:** Identify the powers of x. Analyze the polynomial 2x⁴ + 3x³ + 4x² - 7x - 8. Here, the powers of x are noted as 4, 3, 2, 1, and 0.
• **Step 2:** Determine the order of powers. Look at how these powers are sequenced: 4, 3, 2, 1, 0. Notice that the powers decrease from left to right. This sequence confirms that the polynomial is written in decreasing order.
• **Step 3:** Conclusion on order type. Since the powers visibly decline from 4 down to 0, the answer is that the polynomial is in decreasing order.

This systematic approach enables you to break down a problem into bite-sized pieces, making it much more manageable and easier to comprehend, particularly in a learning environment.