Problem 59 - Algebra A Combined Function

Question

Write each polynomial in descending powers of the variable and with no missing powers. See Example 15. $$ 7 x^{2}+3 $$

Step-by-Step Solution

Answer

The polynomial is \(7x^{2} + 0x + 3\) in descending order with no missing terms.

Step 1: Identify the Terms

The given polynomial is \(7x^{2} + 3\). It consists of two terms: \(7x^{2}\) (a quadratic term) and \(3\) (a constant term).

Step 2: Arrange Polynomial by Powers of x

In descending order of powers of \(x\), we list the terms. Since \(x^2\) is already the highest degree term and \(3\) is the constant term, the polynomial is already in descending order: \(7x^{2} + 3\). There are no other powers of \(x\) to include.

Step 3: Address Any Missing Terms

In a complete polynomial, all intermediate powers from the highest to the constant should be represented, even if their coefficients are zero. Since the next lower power after \(x^2\) is \(x^1\), which is missing, we include it as \(0x\). This completes our polynomial as \(7x^{2} + 0x + 3\).

Key Concepts

Descending Powers

When arranging polynomials, we often speak about their terms being in 'descending powers'. This simply means that each term of the polynomial is written in order according to the power of the variable, starting with the highest and ending with the lowest. For example, in the polynomial \(7x^2 + 3\), the terms are already provided in descending powers because the term with the highest power, \(7x^2\), is listed first, followed by the constant term \(3\), which is equivalent to \(3x^0\).
As you arrange polynomials, always look for the highest power present, and format the terms from there. This practice simplifies comparison of polynomials and solving polynomial equations.

Missing Terms

Missing terms in a polynomial are those intermediate power terms which are not explicitly present in the expression. When writing a polynomial fully, you should list all powers of the variable down to the constant, even if some of these have a coefficient of zero.

• For instance, in the polynomial \(7x^2 + 3\), there is no \(x^1\) term, so it can be written as \(7x^2 + 0x + 3\).
• Though we don't always write terms with a zero coefficient, acknowledging their absence helps in certain algebraic manipulations and understanding.

Identifying missing terms is crucial for accurately performing polynomial operations, like addition or subtraction, where aligning terms with the same power is necessary.

Quadratic Term

A quadratic term in a polynomial is one where the variable is raised to the power of two. It's called 'quadratic' because 'quadra-' comes from Latin, meaning square, which relates to the square of the variable.
In our example, \(7x^2\) is the quadratic term. It signals that the polynomial has a degree of 2, making it a quadratic polynomial. Quadratic terms are significant because they determine the highest degree of the polynomial, which affects the shape of related graphs (like parabolas) and solutions to equations.
Understanding the role of a quadratic term is key in many mathematical contexts, such as solving quadratic equations or evaluating expressions involving quadratic polynomials.

Constant Term

A constant term is the part of a polynomial that does not have any variables attached to it. Simply put, it's the number that stands alone in the polynomial expression.
In \(7x^2 + 3\), the \(3\) is the constant term. Regardless of the value of \(x\), the constant term remains unchanged, which is why it's vital as a base reference in evaluating polynomials and graphing.

• Constant terms are also significant as they often represent a starting point or fixed quantity in real-world problems described by polynomials.

Recognizing a constant term's place within a polynomial helps in various operations such as finding a polynomial's value when substituting a number for the variable or in integration.