Problem 17 - Elementary Algebra

Question

Simplify. $$ x 2+2 x 2+3 x-28+x-22 x 2+3 x-28 $$

Step-by-Step Solution

Answer

The expression simplifies to \(x^2 + 7x - 56\).

Step 1: Combine Like Terms

Identify and combine like terms in the expression: \(x^2 + 2x^2 + 3x - 28 + x - 2x^2 + 3x - 28\). Like terms are those terms that have the same variables raised to the same powers.

Step 2: Simplify the Polynomial Terms

First, simplify the quadratic terms: \(x^2 + 2x^2 - 2x^2\). Combine these to get \(x^2\).

Step 3: Simplify the Linear Terms

Next, simplify the linear terms: \(3x + x + 3x\). Combine these to get \(7x\).

Step 4: Simplify the Constant Terms

Finally, simplify the constant terms: \(-28 - 28\). Combine these to get \(-56\).

Step 5: Write the Simplified Expression

Combine the simplified terms to write the final simplified expression: \(x^2 + 7x - 56\).

Key Concepts

Like Terms

When simplifying polynomials, the first thing to do is to identify like terms. Like terms in algebra are terms that have the same variable raised to the same power. This means you can combine or add them together easily.

In the given exercise, we have the expression: \(x^2 + 2x^2 + 3x - 28 + x - 2x^2 + 3x - 28\). Identify and circle the terms with the same variables and powers.

• \(x^2\), \(2x^2\), and \(-2x^2\) are like terms because they all have the variable \(x\) squared.
• \(3x\), \(x\), and \(3x\) are another set of like terms; they have the variable \(x\) to the power of 1.
• -28 and -28 are also like terms, as they are constant terms.

Combining like terms is a principal step in polynomial simplification that simplifies the expression considerably.

Quadratic Terms

Quadratic terms are those that contain a variable raised to the power of two. These are important in algebra because they define the shape of a parabola on a graph.

In our example, the quadratic terms are \(x^2\), \(2x^2\), and \(-2x^2\). To simplify, add or subtract these coefficients and keep the variable and its exponent the same. Here's how it's done:

• Combine: \(x^2 + 2x^2 - 2x^2\).
• The result is: \(x^2\). The coefficients (1, 2, and -2) add up to 1.

Understanding and simplifying quadratic terms helps simplify expressions and solve quadratic equations effectively.

Linear Terms

Linear terms are the elements of an expression that have variables raised to the power of one. They are fundamental for defining straight lines in algebraic equations.

In the expression \(3x + x + 3x\), these are the linear terms, where the variable \(x\) is to the first power. We combine them as follows:

• Add the coefficients: \(3 + 1 + 3\).
• The sum is: \(7x\).

By simplifying these linear terms, you can reduce the complexity of an expression and find solutions much more efficiently. It's like gathering all pieces of a puzzle into a single piece that fits perfectly into your solution.

Constant Terms

Constant terms in a polynomial are numbers without variables. They remain constant—they do not change as the variables in an expression rise and fall.

In the expression, \(-28 - 28\) are constant terms. Simplify them as you would with simple arithmetic:

• Combine these numbers: \(-28 - 28\).
• The sum is: \(-56\).

Even though they stay constant, understanding how to combine constant terms is essential for simplifying any polynomial, leaving you with a more straightforward expression.