Problem 30 - Algebra Form and Function

Question

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ 5-3 x^{7} $$

Step-by-Step Solution

Answer

Question: Write the given polynomial in standard form and find its degree: $$5-3 x^{7}$$ Answer: The polynomial in standard form is $$-3x^7 + 5$$ and its degree is 7.

Step 1: The given polynomial is: $$5 - 3x^7$$ We see two terms here: a constant term '5' and a term with the variable \(x^7\), which has a coefficient of -3. #Step 2: Determine the coefficients of the polynomial#

In the given polynomial, we have: $$a_7 = -3$$ There is also a constant term: $$a_0 = 5$$ Since there are no other terms given, we can assume that the coefficients of all other powers of x are 0. So, $$a_1 = a_2 = ... = a_6 = 0$$ #Step 3: Write the polynomial in standard form#

Step 2: Now that we have determined all the coefficients, we can write the polynomial in standard form: $$-3x^7 + 0x^6 + 0x^5 + 0x^4 + 0x^3 + 0x^2 + 0x^1 + 5$$ However, in a more simplified standard form, we write it as: $$-3x^7 + 5$$ #Step 4: Calculate the degree of the polynomial#

The degree of a polynomial is the highest power of x present in the polynomial. In our case, the highest power of x is 7. So, the degree of the polynomial is 7.

Key Concepts

Polynomial Coefficients

In a polynomial, coefficients are the numbers that multiply the variable part. For example, in the polynomial expression \[ a_n x^n + a_{n-1} x^{n-1} + ext{...} + a_1 x + a_0 \] each \( a_i \) represents a coefficient. These numbers tell us how many units of the variable's power are present. For \(-3x^7 + 5\), the coefficients are the following:

• The coefficient of \(x^7\) is \(-3\).
• The constant term, which can be thought of as \(x^0\), has the coefficient \(5\).
• All other coefficients that are not present explicitly, such as those for \(x^1, x^2, \ldots, x^6\), are understood to be \(0\).

This means our polynomial has coefficients \(a_7 = -3\) and \(a_0 = 5\), with all other coefficients being \(0\). The coefficients give vital information about the terms and structure of the polynomial. They play a crucial role in polynomial evaluation and graphing.

Polynomial Degree

The degree of a polynomial is an important concept that indicates the highest power of the variable present. It reflects the maximum number of solutions or roots the polynomial can have. The degree is a key factor in determining the shape and behavior of the polynomial's graph. For the polynomial \(-3x^7 + 5\), the degree is 7 because the highest power of \(x\) that appears is \(x^7\).

• Remember that the degree is only determined by the highest exponent of \(x\) with a non-zero coefficient.
• The degree can also hint at the end behavior of the polynomial's graph; higher degrees can mean more complex shapes.

Understanding the degree helps us not just to classify the polynomial, but also to predict its behavior under various operations such as differentiation and integration. Overall, determining the degree is one of the first and most crucial steps when analyzing any polynomial expression.

Polynomial Terms

Polynomials are made up of terms, each consisting of a coefficient and a variable part raised to a power. When analyzing or writing a polynomial, it's beneficial to break it into its individual terms. Consider the polynomial \(-3x^7 + 5\):

• It has two terms: \(-3x^7\) and \(5\).
• The term \(-3x^7\) features both a coefficient \(-3\)and a variable raised to the seventh power.
• The term \(5\) is the constant term, which involves no variable part.

Terms can be viewed as building blocks that determine the polynomial's expression. In the standard form, all terms are arranged in descending order based on the power of the variable. Modifying coefficients or exponents in any term alters the polynomial significantly, affecting its graph and solutions. Understanding each term is vital when attempting operations such as addition, subtraction, or even factoring of polynomials.