Perform the indicated operations. See Examples 2–6. (x^4-3x^2+2) - (-2x^4+x^2-3)

Verified step by step guidance

1. Identify the expression to simplify: $(x^4-3x^2+2)-(-2x^4+x^2-3)$.
2. Distribute the negative sign across the second polynomial: $-(-2x^4+x^2-3)$ becomes $+2x^4-x^2+3$.
3. Rewrite the expression by combining the polynomials: $(x^4-3x^2+2)+(2x^4-x^2+3)$.
4. Combine like terms: $x^4+2x^4$, $-3x^2-x^2$, and $2+3$.
5. Simplify each group of like terms to get the final expression.

Key Concepts

Polynomial Operations

Polynomial operations involve performing arithmetic operations such as addition, subtraction, multiplication, and division on polynomial expressions. In this case, we are focusing on subtraction, which requires distributing the negative sign across the second polynomial and then combining like terms. Understanding how to manipulate polynomials is essential for solving algebraic expressions.

Like Terms

Like terms are terms in a polynomial that have the same variable raised to the same power. For example, in the expression x^4 and -2x^4 are like terms because they both contain x raised to the fourth power. Identifying and combining like terms is crucial when simplifying polynomial expressions, as it allows for the reduction of the expression to its simplest form.

Distributive Property

The distributive property states that a(b + c) = ab + ac, allowing us to distribute a factor across terms within parentheses. In the context of polynomial subtraction, this property is applied when subtracting one polynomial from another, as it requires distributing the negative sign to each term of the polynomial being subtracted. Mastery of this property is vital for correctly performing operations on polynomials.