Problem 27 - Elementary Algebra

Question

Simplify. $$ 2(3 x 2-2 x+1)-(5 x-7) $$

Step-by-Step Solution

Answer

The simplified expression is \( 6x^2 - 9x + 9 \).

Step 1: Distribute the Constants

First, distribute the 2 over the terms inside the first set of parentheses: \[ 2(3x^2 - 2x + 1) = 2 \times 3x^2 + 2 \times (-2x) + 2 \times 1 = 6x^2 - 4x + 2 \]

Step 2: Distribute the Negative Sign

Now, distribute the negative sign to the terms inside the second set of parentheses: \[ -(5x - 7) = -5x + 7 \]

Step 3: Combine Like Terms

Combine the results from Steps 1 and 2: Starting with: \[ 6x^2 - 4x + 2 - 5x + 7 \]Combine the \( x \) terms: \[ -4x - 5x = -9x \]Combine the constant terms: \[ 2 + 7 = 9 \]Thus, the expression becomes: \[ 6x^2 - 9x + 9 \]

Step 4: Final Simplified Expression

The final simplified expression is: \[ 6x^2 - 9x + 9 \]

Key Concepts

Distributive Property

The distributive property is a helpful tool when simplifying expressions. It allows you to distribute multiplication over addition or subtraction inside parentheses. For example, if you have an expression like \( a(b + c) \), you can distribute \( a \) across both \( b \) and \( c \), resulting in \( ab + ac \). This property is useful to eliminate parentheses and combine terms more straightforwardly.
In the original exercise, you used the distributive property in two ways:

• Multiplying the number \( 2 \) across the terms inside the parentheses \( 2(3x^2 - 2x + 1) \). After applying the distributive property, each term is multiplied by \( 2 \), transforming the expression into \( 6x^2 - 4x + 2 \).
• Distributing the negative sign across the terms in \(-(5x - 7)\). Here, the negative is applied to each term individually, changing the signs, resulting in \(-5x + 7\).

Using the distributive property is a fundamental step in simplifying expressions, making complex equations easier to manage.

Combining Like Terms

Combining like terms is another crucial step when simplifying expressions, especially after the distributive property has been applied. Like terms are terms that have the same variable raised to the same power. For example, \(2x\) and \(5x\) are like terms because they both have the variable \( x \) raised to the power of 1.
In our step-by-step solution, you combined the following:

• Identified the like terms with variable \( x \) as \(-4x\) and \(-5x\). These terms were combined by simply adding their coefficients, resulting in \(-9x\).
• Constant terms, which are numbers without variables, such as \(2\) and \(7\). Adding these together gives us \(9\).

This process helps in reducing the equation to its simplest form, grouping similar elements, and making it more concise, like transforming \( 6x^2 - 4x + 2 - 5x + 7 \) into \( 6x^2 - 9x + 9 \). By sorting and simplifying, we arrive at the neatest possible expression.

Polynomials

Polynomials are algebraic expressions made up of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. A polynomial can consist of different terms, such as constants, linear terms, quadratic terms, etc.
For instance, the expression \( 6x^2 - 9x + 9 \) is a polynomial with:

• A quadratic term \(6x^2\), which suggests it is a second-degree polynomial.
• A linear term \(-9x\), indicating a first-degree component.
• A constant \(9\), which has no variable associated with it.

Polynomials are pivotal in mathematics as they form the foundation for various operations and higher-level math concepts such as factoring, graphing, and solving polynomial equations. Understanding how to manipulate and rewrite these expressions is vital, as it facilitates solving practical problems in science and engineering.