Problem 31 - Elementary Algebra

Question

\(2 x^{2}+x-9+3 x^{2}-2 x+15\)

Step-by-Step Solution

Answer

The simplified expression is \(5x^2 - x + 6\).

Step 1: Combine Like Terms

Look for terms with the same variable raised to the same power. Combine them by adding their coefficients. In this case, combine the terms with the squared variable (\(2x^2\) and \(3x^2\)) and the linear variable (\(x\) and \(-2x\)). Add the constants separately (\(-9\) and \(15\)).\(2x^2 + 3x^2 + x - 2x - 9 + 15\).

Step 2: Simplify the Combined Terms

Add the coefficients of like terms:For the squared terms: \(2x^2 + 3x^2 = 5x^2\)For the linear terms: \(x - 2x = -x\)For the constants: \(-9 + 15 = 6\)Thus, the expression simplifies to: \(5x^2 - x + 6\).

Key Concepts

Simplifying Algebraic Expressions

To simplify algebraic expressions, it's essential to combine like terms.
Like terms are terms that have the same variable raised to the same power.
For example, in the expression given \[2x^{2}+x-9+3x^{2}-2x+15\], the like terms are:

• \(2x^2\) and \(3x^2\)
• \(x\) and \(-2x\)
• \(-9\) and \(15\)

By adding or subtracting the coefficients of these like terms, we can simplify the expression.
Simplification makes the expression more manageable and easier to understand.

Polynomials

The given expression is an example of a polynomial, which is an algebraic expression with multiple terms.
Each term in a polynomial consists of a coefficient (a numerical factor) and variables raised to whole-number exponents.
The general form of a polynomial is: \[a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\]
where each \(a_i\) is a coefficient and \(x\) is the variable.
Polynomials can have different degrees, with the degree being the highest power of the variable in the expression.
In the simplified expression \[5x^2 - x + 6\], we have:

• A quadratic term \(5x^2\) (degree 2)
• A linear term \(-x\) (degree 1)
• A constant term \(6\) (degree 0)

Knowing the structure of polynomials helps in understanding and simplifying them.

Coefficients

Coefficients are the numerical factors of each term in an algebraic expression.
They can be positive, negative, or zero.
In the expression we're simplifying, the coefficients are:

• \(2\) and \(3\) for the \(x^2\) terms
• \(1\) and \(-2\) for the \(x\) terms
• \(-9\) and \(15\) for the constant terms

To simplify, we combine the coefficients of like terms:

• \(2 + 3 = 5\) for \(x^2\)
• \(1 - 2 = -1\) for \(x\)
• \(-9 + 15 = 6\) for the constants

This process gives us the final simplified expression: \[5x^2 - x + 6\].
Understanding coefficients is key to simplifying any algebraic expression effectively.