Question

For Exercises 49–54, write each polynomial in descending order of degree. $$14 t^{6}+9 t^{3}+5 t^{2}-8 t^{4}-1$$

Short answer

Answer: The polynomial rearranged in descending order of degree is $14t^6 - 8t^4 + 9t^3 + 5t^2 - 1$.

Step-by-step solution

Identify the degrees of each term

Look at each term in the polynomial and note their degrees. In this exercise, the variable is 't'. $$14 t^{6}$$ has a degree of 6. $$9 t^{3}$$ has a degree of 3. $$5 t^{2}$$ has a degree of 2. $$-8 t^{4}$$ has a degree of 4. $$-1$$ has a degree of 0. (since it's a constant)

Rearrange the terms in descending order of degree

Now that we have the degrees of each term, we can reorder the terms such that they are arranged in descending order of degree, i.e., from highest degree to lowest degree. So the rearranged polynomial becomes: $$14 t^{6} - 8 t^{4} + 9 t^{3} + 5 t^{2} - 1$$ The given polynomial in descending order of degree is: $$14 t^{6} - 8 t^{4} + 9 t^{3} + 5 t^{2} - 1$$

Key concepts

Polynomial Terms

In a polynomial, each part separated by a plus \((+)\) or minus \((-\) sign is known as a polynomial term. The complete expression consists of one or more terms that are connected together. Each term includes:

• A coefficient, which is usually a number.
• A variable, such as "\(t\)" in our example, which can have different values.
• An exponent that dictates how many times the variable is multiplied by itself.

For instance, in the term \(14t^6\), "\(14\)" is the coefficient, "\(t\)" is the variable, and "\(^6\)" is the exponent. Recognizing each term in a polynomial and understanding what each component represents is foundational in working with polynomials. Writing each polynomial distinctly helps in identifying them and performing operations accurately.

Degree of a Polynomial

The degree of a term is determined by its exponent of the variable. With multiple terms, the degree of the entire polynomial is the highest degree present among those terms.
For example, in the polynomial \(14 t^{6} + 9 t^{3} + 5 t^{2} - 8 t^{4} - 1\), we have the following degrees for each term:

• \(14 t^{6}\) has a degree of 6.
• \(9 t^{3}\) has a degree of 3.
• \(5 t^{2}\) has a degree of 2.
• \(-8 t^{4}\) has a degree of 4.
• \(-1\), being a constant, has a degree of 0.

The degree of this polynomial is therefore 6, which is from the term \(14t^6\). Understanding the degree is crucial as it helps in categorizing polynomials and predicting their behavior when graphed.

Descending Order

Arranging terms in a polynomial involves writing them in a sequence based on their degree, from highest to lowest. This is termed as "descending order" of the degrees.
In practice, you look for the term with the largest exponent and write it first, continue this process until you reorder all terms,normally ending with the constant term as it has the degree 0.For the given polynomial \(14 t^{6} + 9 t^{3} + 5 t^{2} - 8 t^{4} - 1\), the steps involve:

• Identify the term with the highest degree, which is \(14 t^{6}\).
• Next, compare the other terms:
• -\(8 t^{4}\)
• \(9 t^{3}\)
• \(5 t^{2}\)
• Finally, place the constant term \(-1\).

After reordering, the polynomial in descending order becomes \(14 t^{6} - 8 t^{4} + 9 t^{3} + 5 t^{2} - 1\). Writing polynomials in descending order makes them easier to read and paves the way for standardized mathematical operations.