Problem 3 - Algebra Form and Function

Question

Write the polynomials in standard form. $$ x(x-2)+x^{2}(3-x) $$

Step-by-Step Solution

Answer

Question: Write the given polynomial expression in standard form: $x(x-2) + x^2(3-x)$ Answer: The polynomial in standard form is $-x^3 + 4x^2 - 2x$.

Step 1: Distribute the terms within the parentheses

Multiply the terms within the parentheses by the terms outside the parentheses: $$ x(x-2) + x^2(3-x) = x^2 - 2x + 3x^2 - x^3 $$

Step 2: Collect the like terms

Combine the terms with the same power of x: $$ x^2 - 2x + 3x^2 - x^3 = -x^3 + 4x^2 -2x $$

Step 3: Order the terms in descending powers of x

Arrange the terms from the highest to the lowest power of x: $$ -x^3 + 4x^2 - 2x $$ So, the polynomial in standard form is: $$ -x^3 + 4x^2 - 2x $$

Key Concepts

Standard Form

When working with polynomials, the standard form is an important concept. A polynomial is considered to be in standard form when its terms are written in descending order, based on the power of the variable. This means that the term with the highest exponent comes first, followed by the terms with lower exponents, all the way down to the constant term if there is one.
This simply helps us to organize and easily read the expression. For example, given a polynomial such as \(-x^3 + 4x^2 - 2x\), it is in standard form because the terms follow the order of their exponents: from \(x^3\) to \(x^2\) to \(x\).

• Always start by arranging the terms from the highest to lowest exponent.
• This helps in clear communication, as this format is widely understood in mathematical notations.
• Make sure any constant terms, that are exponent zero, are placed at the end.

This format does not affect the value of the polynomial; it remains the same mathematically.

Combining Like Terms

Involving like terms is inevitable when simplifying expressions or polynomials. Like terms are terms that include the same variables raised to the same power. They can be combined because they represent the same entity in different quantities. For example, in the expression \(x^2 - 2x + 3x^2 - x^3\), the like terms are \(x^2\) and \(3x^2\).

• Identify terms with the same variables and exponents: \(x^2\) with \(3x^2\).
• Combine them by simply adding or subtracting their coefficients. In this case: \(x^2 + 3x^2 = 4x^2\).
• This makes the expression simpler and easier to handle. The reduced expression is \(-x^3 + 4x^2 - 2x\).

Remember that only coefficients can be modified when you are combining like terms, as the variables and their exponents remain the same.

Distributing Terms

Distributing is a process that allows us to eliminate parentheses by multiplying each term inside the parentheses by the term outside. This is essential in simplifying the polynomial expressions effectively. It is akin to an application of the distributive property which states that \(a(b+c) = ab + ac\).

Taking the polynomial \(x(x-2) + x^2(3-x)\), we use distribution to simplify this expression:

• For the first term \(x(x-2)\), multiply \(x\) by each term inside the parentheses: \(x \cdot x - x \cdot 2 = x^2 - 2x\).
• Similarly, for the second \(x^2(3-x)\), distribute \(x^2\): \(x^2 \cdot 3 - x^2 \cdot x = 3x^2 - x^3\).
• This process removes the parentheses and expresses the polynomial entirely in terms of individual algebraic terms, making it \(x^2 - 2x + 3x^2 - x^3\).

Always distribute carefully to ensure you are multiplying each component accurately. Double-checking your steps can help you avoid errors commonly made in calculations.