Problem 78 - Elementary Algebra

Question

Simplify. $$ 6(-2 x 2+3 x-1)+10 x 2-5 x $$

Step-by-Step Solution

Answer

Simplified expression: \(-2x^2 + 13x - 6\).

Step 1: Distribute the 6

Start by distributing the 6 across each term inside the parentheses. Calculate separately for each term in the expression: - For \(-2x^2\): \[6 imes (-2x^2) = -12x^2\] - For \(3x\):\[6 imes 3x = 18x\] - For \(-1\):\[6 imes (-1) = -6\]This makes the expression: \[-12x^2 + 18x - 6 +10x^2 - 5x\]

Step 2: Combine Like Terms

Now, combine the terms that contain the same powers of \(x\):- Combine the \(x^2\) terms: \(-12x^2 + 10x^2 = -2x^2\)- Combine the \(x\) terms: \(18x - 5x = 13x\)Thus the expression simplifies to:\[-2x^2 + 13x - 6\]

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that is used to simplify expressions. It involves distributing, or passing out, a number multiplying a group of numbers or terms inside parentheses across each element within those parentheses. For example, if you have an expression like \( a(b + c) \), using the distributive property, it becomes \( ab + ac \). This helps in breaking down and simplifying algebraic expressions.

In our exercise, the distributive property was applied to the expression \( 6(-2x^2 + 3x - 1) \). Each term inside was multiplied by the number outside the parentheses (which is 6 in this case), resulting in:

• \( 6 \times (-2x^2) = -12x^2\)
• \( 6 \times 3x = 18x\)
• \( 6 \times (-1) = -6\)

This breaks down the expression into separate terms, making it more manageable. The distributive property can be applied beyond mere numbers and used in various algebraic contexts, such as with polynomials, which we will encounter later.

Combining Like Terms

Combining like terms is an essential process in algebra that helps to simplify expressions further by merging terms that have the same variables raised to the same power. Simply put, it is gathering all similar terms together to condense an expression into its simplest form.

Like terms refer to terms that contain the same variable parts. For example, \( 3x \) and \( 5x \) are like terms because they both have \( x \). Similarly, \( 7x^2 \) and \( -2x^2 \) are like terms because they share the \( x^2 \).

In the solution to our exercise, the expression \(-12x^2 + 18x - 6 + 10x^2 - 5x\) was simplified by combining the like terms:

• \(-12x^2 + 10x^2 = -2x^2\)
• \(18x - 5x = 13x\)

All the other terms did not have any comparable variables or powers, so they remained the same. By combining like terms, the expression was reduced to \(-2x^2 + 13x - 6\), showcasing the simplicity that can be achieved.

Polynomials

Polynomials are algebraic expressions that consist of variables and coefficients, structured as a sum of terms. Each term in a polynomial is composed of a coefficient (a number), a variable (such as \( x \)), and a non-negative integer exponent. For example, \( 7x^2 - 3x + 4 \) is a polynomial with three terms: \( 7x^2 \), \(-3x\), and \( 4 \).

In the exercise, the expression was a polynomial involving powers of \( x \) up to 2, given by the terms \(-2x^2\), \(13x\), and \(-6\). Polynomials are versatile in mathematics and appear in numerous concepts like equations, functions, and calculus. They can be classified by their degree - the highest power of the variable present. In our example, the degree is 2, making it a quadratic polynomial.

Polynomials can be manipulated using various algebraic operations such as addition, subtraction, multiplication (using the distributive property), and division. Understanding polynomials provides a foundation for advanced topics and problem-solving techniques in algebra.