Question

consider the following polynomial:
\\6x + 5x^4 - 10x + 8x^3 - 7x^2 + 9^2\\

written in standard form, the polynomial has \\(\text{______}\\) terms.
the constant term is \\(\text{______}\\).
in standard form, the coefficient of the linear term is \\(\text{______}\\).

Explanation:

🆕 New Concept Discovered: Polynomials in Standard Form
Ordering terms from highest degree to lowest

Step 1: Combine like terms

To write the polynomial in standard form, we first simplify it by combining any like terms.

Given polynomial:
\[ 6x + 5x^4 - 10x + 8x^3 - 7x^2 + 9^2 \]

Identify the like terms:

• The terms with \( x \) are \( 6x \) and \( -10x \).
• Combine them: \( 6x - 10x = -4x \).

Simplify the constant term:

• \( 9^2 = 81 \)

The simplified terms are:
\[ 5x^4, \quad 8x^3, \quad -7x^2, \quad -4x, \quad 81 \]

Step 2: Write in standard form

Standard form means writing the terms in descending order of their exponents (from highest to lowest degree).

\[ 5x^4 + 8x^3 - 7x^2 - 4x + 81 \]

Step 3: Identify the polynomial features

• Number of terms: Count the distinct terms separated by addition or subtraction in the simplified standard form:

\[ 5x^4, \quad 8x^3, \quad -7x^2, \quad -4x, \quad 81 \]
There are \( 5 \) terms.

• Constant term: The term without any variable \( x \):

\[ 81 \]

• Coefficient of the linear term: The linear term is the term with \( x^1 \) (or simply \( x \)), which is \( -4x \). Its coefficient is:

\[ -4 \]

Answer:

• Written in standard form, the polynomial has 5 terms.
• The constant term is 81.
• In standard form, the coefficient of the linear term is -4.