Problem 60 - Algebra A Combined Function

Question

Write each polynomial in descending powers of the variable and with no missing powers. See Example 15. $$ 5 x^{2}-2 $$

Step-by-Step Solution

Answer

The polynomial is \( 5x^2 + 0x - 2 \).

Step 1: Identify Powers of the Variable

In the polynomial \( 5x^2 - 2 \), the variable \( x \) is raised to the power of 2, which is the highest power. We need to ensure the polynomial reflects descending order, starting with this highest power and going to the lowest.

Step 2: Determine Missing Terms

Identify any missing powers of the variable \( x \). In this case, there is no \( x^1 \) term or \( x^0 \) term explicitly written out (\( x^0 \) corresponds to the constant term -2). We should include an \( x^1 \) term even if its coefficient is zero.

Step 3: Rewrite with All Powers Represented

Rewrite the polynomial by inserting the missing \( x \) terms with a coefficient of zero: \( 5x^2 + 0x - 2 \). This ensures all powers from 2 (highest) to 0 are represented.

Step 4: Verify Descending Order

Check that the polynomial now shows all possible powers of \( x \) in descending order: \( x^2, x^1, x^0 \). The polynomial \( 5x^2 + 0x - 2 \) is in the correct format.

Key Concepts

Descending Powers

In polynomials, arranging terms in descending powers means listing them from the highest power of the variable down to the lowest. This is a standard method for writing polynomials, making them easier to read and manipulate. When you look at a polynomial, you'll generally see that it starts with the term that has the largest exponent and ends with the constant term (if any), which is the term with the variable raised to the power of zero.

• The term with the variable having the highest exponent is written first.
• The next highest exponent terms follow in order until you reach the constant term.

For example, in the polynomial given - before rewriting it as \[5x^2 - 2\]the variable \( x \) is raised to the power of 2, making it the highest. Hence, it comes first in the expression.

Missing Terms

When a polynomial is given, sometimes certain powers of the variable may not appear explicitly. These are known as missing terms. Especially when organizing a polynomial in descending powers, it's essential to recognize if any intermediate powers are missing and represent them even if their coefficient is zero.

• Identify which variable powers are present and which are missing.
• Add missing terms with zero coefficients to maintain completeness.

In the exercise example, after identifying all possible powers of \( x \) from 2 down to 0, it's clear that the \( x^1 \) term is missing. It is important to insert \( 0x \)to ensure completeness, giving every power a place, even if it does not change the polynomial's value.

Variable Powers

Variable powers are the exponents of the variable in each term of a polynomial. Each power is a distinct degree of \( x \), affecting the term's contribution to the polynomial's shape on a graph. Understanding variable powers helps in organizing the polynomial and solving or evaluating it.

• Variable powers are represented as exponents in each term.
• Higher powers indicate stronger influence on the polynomial's behavior at larger values of the variable.

In our example, the highest variable power is \( 2 \) given by \( x^2 \). Even though an \( x^1 \) term isn't visibly there, acknowledging its absence by including a \( 0x \)term ensures each power is properly accounted for.

Rewriting Polynomials

Rewriting polynomials involves arranging all terms to reflect a complete sequence of descending powers, highlighting any previously missing powers with a coefficient of zero. This formalizes the expression for easier evaluation and comparison.

• Start with the highest power and proceed to the lowest.
• Insert zero-coefficient terms for any missing variable power.

For the example polynomial, after noting the terms, you rewrite it as \(5x^2 + 0x - 2\).By doing this, you ensure the polynomial is cohesive and correctly formatted. Ensuring nothing is omitted aids in error-checking when solving or simplifying polynomials, preserving consistency.