Problem 4 - Algebra Form and Function

Question

Write the polynomials in standard form. $$ \frac{x^{4}-2 x-14 x^{3}}{7} $$

Step-by-Step Solution

Answer

Question: Write the given polynomial in standard form: $$\frac{x^{4}-2 x-14 x^{3}}{7}$$ Answer: $$\frac{1}{7}x^4 - 2x^3 - \frac{2}{7}x$$

Step 1: Identify the terms in the polynomial

The polynomial given is: $$ \frac{x^{4}-2 x-14 x^{3}}{7} $$ In this expression, we have three terms: \(x^{4}\), \(-2x\), and \(-14x^{3}\).

Step 2: Divide each term by the given divisor

Now we need to divide each term by the divisor 7: $$ \frac{x^4}{7} - \frac{2x}{7} - \frac{14x^3}{7} $$

Step 3: Simplify the expression

Now we simplify the expression: $$ \frac{1}{7}x^4 - \frac{2}{7}x - 2x^3 $$

Step 4: Write the polynomial in standard form

Now that we have simplified the expression, we can write the polynomial in standard form by ordering the terms based on their exponents from the greatest to the smallest: $$ \frac{1}{7}x^4 - 2x^3 - \frac{2}{7}x $$ This is the polynomial in standard form.

Key Concepts

Polynomial Division

Polynomial division is a process that involves dividing a polynomial by a number or another polynomial. It requires dividing each term of the polynomial individually by the divisor. To understand this, consider the expression \[\frac{x^{4}-2x-14x^{3}}{7}\]. This means we need to divide each term in the numerator by 7.

• Step One: Identify the Terms: Break down the polynomial into its individual terms such as \(x^4\), \(-2x\), and \(-14x^3\).


• Step Two: Divide Each Term: Perform the division separately on each term. The result of dividing \(-14x^3\) by 7 is \(-2x^3\), and so on for the other terms.

After dividing, you get a new expression: \[\frac{1}{7}x^4 - \frac{2}{7}x - 2x^3\]. This subdivision is crucial as it allows simplifying complex polynomials.

Simplifying Expressions

Simplifying expressions is a key step in dealing with polynomials. When you simplify, you aim to write the polynomial in its most reduced form. After dividing, you have \[\frac{1}{7}x^4 - \frac{2}{7}x - 2x^3\]. What you look for in simplification is:

• Reducing Fractions: If any fractional coefficients can be simplified, ensure you do this.


• Combining Like Terms: Although not applicable here, whenever you have terms that can be combined, make sure to do so.


• Ensure Individual Simplification: Each term must stand simplified, like \(-2x^3\) remains clearly broken down from its original form.

It's like cleaning up your polynomial to its lightest form so it becomes much easier to handle and understand.

Polynomial Terms

Understanding polynomial terms is fundamental in both constructing and deconstructing polynomial expressions. Each term comprises a coefficient and one or more variables each with a whole number exponent. For example, in \(-2x^3\), the term consists of:

• Coefficient: This is \( -2 \), which is the number part that multiplies the variable.


• Variable and Exponent: \(x\) is the variable, and \(3\) is the exponent indicating how many times the variable is multiplied by itself.

A polynomial term helps define the structure of the polynomial by contributing to its degree, which is the highest exponent present in the expression. Recognizing terms and organizing them effectively leads to writing polynomials in standard form, such as ordering by descending exponent values, as you see in \[\frac{1}{7}x^4 - 2x^3 - \frac{2}{7}x\]. Understanding each term's role makes handling larger expressions easier and creates clarity.