Problem 11 - Elementary and Intermediate Algebra

Question

Complete each solution. Write the polynomial \(2 x^{2}-1+5 x^{4}\) in descending powers of \(x\) and insert placeholders for each missing term.

Step-by-Step Solution

Answer

Reordered with placeholders: \(5x^4 + 0x^3 + 2x^2 + 0x - 1\).

Step 1: Identify Polynomial Structure

The polynomial given is \(2x^2 - 1 + 5x^4\). Polynomials should be written with terms ordered in descending powers.

Step 2: Arrange Terms by Descending Powers

Reorder the terms so that they are arranged from the highest power of \(x\) to the lowest. The highest exponent is \(4\), followed by \(2\), and the constant term. This gives us \(5x^4 + 2x^2 - 1\).

Step 3: Insert Missing Terms as Zero Placeholder

Since there is no term with \(x^3\) and \(x\), insert placeholders by adding \(0x^3\) and \(0x\) in the polynomial. This brings the polynomial to: \(5x^4 + 0x^3 + 2x^2 + 0x - 1\).

Key Concepts

Descending Powers

When we talk about arranging polynomials in descending powers, it simply means that we organize the terms from the highest exponent to the smallest. Each term in a polynomial is a combination of a coefficient, a variable, and an exponent, such as in the expression \(5x^4\). In our example, we see the polynomial \(2x^2 - 1 + 5x^4\). This is initially unordered. To fix it, we need to rearrange it so that it follows the sequence where the exponents decrease step by step.

- The highest degree term in this expression is \(5x^4\), therefore it should come first.- Then follows any term with the next lower degree, such as \(2x^2\). - Finally, insert any remaining terms like constants, which are equivalent to \(x^0\).

After adjustment, the polynomial will become \(5x^4 + 2x^2 - 1\). This order helps in analyzing and working further with polynomials easier.

Missing Terms

Missing terms in a polynomial can confuse students if they aren't explicitly stated. When writing polynomials, especially for computational purposes, it's essential to represent all degrees, even those with zero coefficients. If a degree is missing, like \(x^3\) and \(x^1\) in our example \(5x^4 + 2x^2 - 1\), it means they have a coefficient of zero.

Here's why it's helpful:

• It keeps consistency in the sequence of descending powers.
• It simplifies operations like addition, subtraction, and finding derivatives.
• It provides a comprehensive view of all degrees present, avoiding missed calculations.

Therefore, understanding and identifying these missing elements are crucial in polynomial manipulation.

Placeholder Terms

Inserting placeholders involves adding terms to represent degrees that are not present in the polynomial. You achieve this by including zero coefficients for any missing degrees. In the polynomial \(5x^4 + 2x^2 - 1\), placeholders are used for the missing \(x^3\) and \(x^1\) terms.

Incorporating placeholder terms means writing:

• \(0x^3\) to indicate there is no cubic term.
• \(0x\) to show a missing linear term.

These placeholders modify the polynomial to \(5x^4 + 0x^3 + 2x^2 + 0x - 1\). Even though they evaluate to zero and don't change the polynomial's value, placeholders maintain format consistency, which is useful for performing polynomial arithmetic. Recognizing and using placeholder terms appropriately ensures that you can transition smoothly between different polynomial expressions.