Problem 43 - Elementary Algebra

Question

Simplify. $$ x 3 x+2 x-2+43 x(x-2) $$

Step-by-Step Solution

Answer

The simplified expression is \( 46x^2 - 84x - 2 \).

Step 1: Distribute and Simplify Terms

The given expression is \( x \, 3x + 2x - 2 + 43x(x-2) \). We will first simplify the terms by expanding and combining like terms. Split the expression by multiplying where needed:\[ x \cdot 3x + 2x - 2 + 43x(x-2) \] which simplifies to \[ 3x^2 + 2x - 2 + 43x^2 - 86x \].

Step 2: Combine Like Terms

Now, collect and combine like terms from the simplified expression: \( 3x^2 + 2x - 2 + 43x^2 - 86x \). Combine the \( x^2 \) terms and the \( x \) terms separately.\[ (3x^2 + 43x^2) + (2x - 86x) - 2 \].

Step 3: Complete the Simplification

Add and subtract the like terms:\[ 46x^2 - 84x - 2 \]. Now the expression is fully simplified.

Key Concepts

Distributive Property

The Distributive Property is a fundamental algebraic concept used to simplify expressions. It's like spreading elements across a sum or difference within parentheses. Imagine you have an expression like \(a(b+c)\). You apply the Distributive Property by multiplying each term inside the parentheses by the term outside. So, \(a(b+c)\) becomes \(ab + ac\).
In the exercise given, when we distribute \(43x\) over \((x-2)\), we follow this logic, resulting in \(43x \cdot x + 43x \cdot (-2)\), which expands into \(43x^2 - 86x\).
This step is essential in building the foundation for further simplification tasks. Distributing terms correctly helps in organizing and reducing complex expressions.

Combining Like Terms

Combining like terms is an important simplification technique in algebra. It requires identifying terms in an expression that have exactly the same variables with the same exponents. Once identified, these terms can be added or subtracted together to lessen the complexity of the expression.
For instance, consider terms like \(3x^2\) and \(43x^2\). These are 'like terms' because both are forms of \(x^2\). Hence, you can combine them to become \(46x^2\).
Similarly, for linear terms like \(2x\) and \(-86x\), you combine them to obtain \(-84x\).

• Always check for like terms that have the exact same variable and exponent.
• Try to simplify by adding or subtracting the coefficients.
• Ordering the terms often helps in easily spotting the similar terms for combining.

Effective combining of like terms reduces the expression making it more efficient to work with.

Polynomial Expansion

Polynomial expansion involves expressing a product of polynomials as a sum of terms. This often requires the use of both distribution and combining like terms, as previously described.
In the presented exercise, one polynomial component is \(43x(x-2)\). Here, polynomial expansion transforms it into an expanded form as part of simplifying the expression: \(43x^2 - 86x\).
This technique allows the conversion of a polynomial product into an easier-to-handle sum of individual terms. It's an essential procedure, particularly when simplifying expressions before solving more complex algebraic equations.
By expanding, each part of the polynomial is broken down, allowing a clearer path to simplification. Remember to distribute each term over the entire polynomial, not skipping any intermediate steps.