Problem 18 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation(s) and write the resulting polynomial in standard form.\(\left(3 x^{2}+1\right)-\left(2 x^{2}-2 x+3\right)\)

Step-by-Step Solution

Answer

The resulting polynomial in standard form after performing the indicated operation is \(x^2 + 2x - 2\).

Step 1: Identify Like Terms

In the two polynomials \(3x^2 + 1\) and \(2x^2 - 2x + 3\), the like terms are \(3x^2\) from the first polynomial and \(2x^2\) from the second polynomial, and the constant terms '1' from the first polynomial and '3' from the second polynomial.

Step 2: Subtract Like Terms

Subtract the like terms from the first polynomial and the second polynomial. One has to be careful with the signs here. The subtraction yields \(3x^2 - 2x^2\) for \(x^2\) terms and \(1 - 3\) for constant terms.

Step 3: Simplify the Resulting Polynomial

On simplifying, \(3x^2 - 2x^2 = x^2\) and \(1 - 3 = -2\). The term \(-2x\) from the second polynomial doesn't have a corresponding term in the first polynomial, so it remains as it is but changes the sign. Therefore, the resulting polynomial is \(x^2 + 2x - 2\).

Key Concepts

Like Terms

Understanding like terms is essential when working with polynomials. Like terms in algebra are terms that have exactly the same variable part. That means they have identical variables raised to the same power (or exponent).
For example:

• In the expressions \(3x^2\) and \(2x^2\), the terms are like terms because both include \(x^2\).
• However, in the expressions \(3x^2\) and \(2x\), the terms are not like terms, as the variable parts differ.

When combining like terms, you adjust only the coefficients (the numbers in front of the variabless), not the variable part.
This is helpful in polynomial subtraction, as you can only directly subtract or add like terms.

Constant Terms

Constant terms are just that—constant. They do not change or include any variable. In other words, they are numbers without a variable part, including zero-degree terms.
For instance, in the polynomials given:

• \(3x^2 + 1\) contains the constant term \(1\).
• \(2x^2 - 2x + 3\) contains the constant term \(3\).

When performing subtraction or addition of polynomials, it’s crucial to correctly identify and handle constant terms separately.
They behave like terms when solution steps involve them. For example, subtracting constant terms \(1 - 3\) results in \(-2\).
Handling constant terms appropriately is critical in simplifying any polynomial.

Simplifying Polynomials

Simplifying polynomials involves decreasing the complexity of the expressions by combining like terms and arranging the results in standard form. The standard form is where the polynomial is ordered in descending power of the variables.
Consider the given exercise:

• Start by identifying and subtracting like and constant terms.
  The subtraction of \(3x^2 - 2x^2\) provides \(x^2\) as a simplified expression.
• Also process the constant terms: \(1 - 3\) results in \(-2\).
• If any term (like \(-2x\)) cannot be directly subtracted due to absence of a like term, it stands unchanged in the simplified result.

Thus, the final polynomial after simplification becomes \(x^2 + 2x - 2\).
Learning to simplify polynomials correctly helps students solve polynomial equations more efficiently and understand complex algebraic expressions better.