Problem 64 - Algebra 1

Question

COMBINING LIKE TERMS Apply the distributive property. Then simplify by combining like terms. $$ (5-2 x)(-x)+x^{2} $$

Step-by-Step Solution

Answer

The simplified expression after applying the distributive property and combining like terms is \(3x^2 - 5x\).

Step 1: Apply Distributive Property

Firstly, apply the distributive property to the given problem. The distribution is between '-x' and '5-2x'. So, each term in the parenthesis will be multiplied by '-x'. This gives: \[ (-x)(5) + (-x)(-2x) + x^2 \]. When simplified, this becomes \[ -5x + 2x^2 + x^2 \].

Step 2: Combine Like Terms

Next, combine the like terms. Here, like terms are those terms that have the same variables and powers. Our like terms are \(2x^2\) and \(x^2\). When we add these terms together, they combine to form \(3x^2\), and the other remaining term is \(-5x\). This means our simplified expression is: \[3x^2 - 5x\].

Key Concepts

Distributive Property

The distributive property is a handy tool in algebra that allows us to simplify expressions. It tells us how to multiply a term on the outside of a parenthesis with each term inside. In mathematical terms, given an expression like \( a(b + c) \), you apply the distributive property by multiplying \( a \) with \( b \) and \( c \):

• \( a \times b \)
• \( a \times c \)

This results in \( ab + ac \).
In our exercise, we used the distributive property to multiply \(-x\) with each term in \(5 - 2x\):

• \((-x)(5) = -5x\)
• \((-x)(-2x) = 2x^2\)

These steps help transform the expression into something more workable, setting the stage for the next step, simplifying expressions.

Simplifying Expressions

Simplifying expressions is the process of making mathematical expressions as straightforward as possible. This often involves performing operations like addition, subtraction, and combining like terms until you reach the simplest form.
After applying the distributive property, the expression from our exercise became \(-5x + 2x^2 + x^2\).
Simplifying involves looking for like terms, which are terms that share the same variables raised to the same powers.
Here:

• \(2x^2\) and \(x^2\) are like terms.
• They combine to form \(3x^2\).

The simplified expression is then written as \(3x^2 - 5x\). Each step reduces complexity, making it easier to solve or use the expression in further calculations.

Polynomial Operations

Polynomial operations involve performing mathematical actions such as addition, subtraction, multiplication, and sometimes division on polynomials. A polynomial is simply an expression made of variables raised to various powers and coefficients. When we engage in polynomial operations, we often apply rules like the distributive property and combining like terms.
In the exercise, the expression began as a polynomial due to the involvement of terms like \(-x\) and \(5 - 2x\). These are manipulated through the use of the distributive property and further operations to reach a simplified form.
Key steps in polynomial operations for our exercise included:

• Distributing \(-x\) across \(5 - 2x\).
• Combining the generated like terms \(2x^2 + x^2\) into \(3x^2\).

Understanding polynomial operations helps in solving a wide range of problems in algebra, making these expressions less intimidating and more manageable.