Problem 65 - Elementary Algebra

Question

Simplify. $$ 1 x+3-x x 2-6 x+9+3 x 2-9 $$

Step-by-Step Solution

Answer

The expression simplifies to \(2x^2 - 5x + 3\).

Step 1: Identify Like Terms

The expression is:\[1x + 3 - xx 2 - 6x + 9 + 3x^2 - 9\]First, identify the like terms which are the variable terms \(x\), \(xx\), \(x^2\), and the constant terms.

Step 2: Correct Writing Errors in the Expression

The expression seems to have a typo. Assuming "xx 2" means \(x^2\), let's rewrite the expression as:\[1x + 3 - x^2 - 6x + 9 + 3x^2 - 9\].

Step 3: Group Like Terms

Group the terms with the same variables. The linear \(x\) terms are \(1x\) and \(-6x\). The quadratic \(x^2\) terms are \(-x^2\) and \(+3x^2\). The constants are \(+3\), \(+9\), and \(-9\).

Step 4: Combine Like Terms

Combine the grouped terms:- \(x - 6x = -5x\)- \(-x^2 + 3x^2 = 2x^2\)- \(3 + 9 - 9 = 3\)This makes the expression \(2x^2 - 5x + 3\).

Step 5: Write the Simplified Expression

All the like terms have been combined resulting in the simplified expression:\[2x^2 - 5x + 3\].

Key Concepts

Like Terms

When simplifying algebraic expressions, the concept of "like terms" is crucial. Like terms are terms that have the same variables raised to the same powers. In the expression given, such as in \[1x + 3 - x^2 - 6x + 9 + 3x^2 - 9,\] the terms with the variable \( x \) are like terms if they have the same power of \( x \).

• For example, \( 1x \) and \( -6x \) are like terms because both are linear, meaning they have \( x^1 \).
• Similarly, \( -x^2 \) and \( +3x^2 \) are like terms, but they belong to the quadratic category because of the \( x^2 \).
• Constants like \(+3\), \(+9\), and \(-9\) are also like terms but without variables.

Recognizing like terms is the first step in simplifying because it helps us understand which terms can be combined. This is essential for simplifying polynomial expressions effectively.

Combining Like Terms

After identifying like terms, the next step is to combine them. This process involves adding or subtracting the coefficients of like terms. Let’s break this down further:

• For the linear terms \( 1x \) and \(-6x\), combining them means calculating \(1 - 6\), which results in \(-5x\).
• For the quadratic terms \(-x^2\) and \(3x^2\), it involves adding the coefficients \(-1 + 3\), giving \(2x^2\).
• The constants \(+3\), \(+9\), and \(-9\) are added together, resulting in \(+3\).

By combining like terms, we simplify the expression into a simpler form. The final simplified form of our expression is \[2x^2 - 5x + 3.\] This method reduces the expression to one that is easier to understand and use in further mathematical operations.

Polynomial Expressions

A polynomial expression consists of variables and constants, involving only operations of addition, subtraction, multiplication, and non-negative integer exponents. Our expression, once simplified to \[2x^2 - 5x + 3,\] demonstrates this perfectly. Here's what makes it a polynomial:

• It includes terms with the variable \( x \) raised to different whole number powers like \( x^2 \) (quadratic) and \( x \) (linear).
• No variable is in the denominator, and there are no fractional or negative exponents.
• The constants, without variables, stand alone and still form part of the polynomial as a whole.

Polynomials can come in various shapes and degrees, and this particular one is called a quadratic polynomial because its highest degree term is \( x^2 \). Understanding polynomial expressions is key in algebra as they form the foundation for many advanced topics.