Question

Write the polynomial \(6 x-2 x^3+x^4-7 x^2+\) 5 in descending order. Write your answer in the blank.

Short answer

The polynomial in descending order is: \(x^4 - 2x^3 - 7x^2 + 6x + 5\)

Step-by-step solution

Identify the terms in the polynomial

The given polynomial is: \[6x - 2x^3 + x^4 - 7x^2 + 5\]Identify each term along with their coefficients and powers of \(x\).

Arrange terms based on the power of \(x\)

Arrange each term in the polynomial based on the descending powers of \(x\). The order should start from the highest power of \(x\) to the lowest (including the constant term, which is considered as \(x^0\)).

Rewrite the polynomial in descending order

Rearrange the terms to get:\[x^4 - 2x^3 - 7x^2 + 6x + 5\]This is the polynomial written in descending order.

Key concepts

polynomials

Polynomials are fundamental expressions in algebra that consist of variables raised to varying powers and multiplied by coefficients, then summed together. Each term in a polynomial is a product of a constant coefficient and a variable raised to an integer power. For instance, in the polynomial \[6x - 2x^3 + x^4 - 7x^2 + 5 \] identifying each term is crucial:

• \( 6x\) - The coefficient is 6 and the variable is raised to the power of 1
• \( -2x^3\) - The coefficient is -2 and the variable is raised to the power of 3
• \( x^4\) - The coefficient is 1 and the variable is raised to the power of 4
• \( -7x^2\) - The coefficient is -7 and the variable is raised to the power of 2
• \( 5\) - This is a constant term with a variable power of 0

Recognizing these terms enables better understanding and manipulation of polynomials in various mathematical contexts.

descending order

Arranging a polynomial in descending order involves ordering the terms from highest to lowest powers of the variable. This systematic arrangement makes it easier to perform operations like addition, subtraction, and differentiation. In our example, we start with the term having the highest power:

• First, we identify \( x^4 \), which is the term with the highest power.
• Next, we look for the term with the next highest power, which is \( -2x^3 \).
• This is followed by \( -7x^2 \), \( 6x \), and finally \( 5 \), the constant term.

Reordering these terms, we get the descending ordered polynomial: \[ x^4 - 2x^3 - 7x^2 + 6x + 5 \]. Rearranging them according to power ensures clarity and consistency.

coefficients

Coefficients are the numerical multipliers of the variable terms in a polynomial. They play a critical role in determining the shape and position of polynomial graphs. Each term in a polynomial consists of a coefficient and a variable raised to a power. For instance, in \[ 6x - 2x^3 + x^4 - 7x^2 + 5 \]

• \(6\) is the coefficient of \( x \).
• \(-2\) is the coefficient of \( x^3 \), indicating that this term will contribute negatively to the overall polynomial.
• \( 1 \) is the implicit coefficient of \( x^4 \), typically represented just as \( x^4 \).
• \(-7\) is the coefficient of \( x^2 \).
• \( 5 \) is the constant term, not associated with any variable.

Understanding coefficients helps in performing polynomial operations and in comprehending how each term affects the overall expression.