Subtracting Polynomials - Algebra-1

1. Fundamental Concepts

Essence of polynomial subtraction: Polynomial subtraction is equivalent to "adding the opposite of the polynomial being subtracted" (Subtract = add a negative). This means converting the subtraction operation into an addition operation, with the core being to change the sign of each term in the polynomial being subtracted and then perform the addition operation with the minuend polynomial.
Definition of like terms: Terms that have the same variables and the same exponents for corresponding variables are called like terms (e.g., \(x^3\) and \(-4x^3\), \(3x^2\) and \(-2x^2\)). Only like terms can be combined in polynomial subtraction.

2. Key Concepts

Rule for sign transformation: Every term in the polynomial being subtracted must have its sign changed (positive to negative, negative to positive), and no term should be omitted. For example, when calculating \((x^3 + 3x^2) - (4x^3 + 2x^2)\), it needs to be converted to \(x^3 + 3x^2 - 4x^3 - 2x^2\).

Rule for combining like terms: When combining like terms, only the coefficients are added or subtracted, while the variables and their exponents remain unchanged. For instance, \(x^3 - 4x^3 = (1 - 4)x^3 = -3x^3\) and \(3x^2 - 2x^2 = (3 - 2)x^2 = x^2\).
Order of operations: First, perform the sign transformation, then identify all like terms, and finally combine the like terms to obtain the simplified polynomial (the result is usually arranged in descending or ascending powers of a certain variable, with descending powers being the common choice).

3. Examples

(1) Easy Difficulty

Calculate \((2x + 5) - (x + 3)\)

Change signs: Change the sign of the subtrahend \((x + 3)\) to \(-x - 3\);
Convert to addition: \(2x + 5 - x - 3\);
Combine like terms: \((2x - x) + (5 - 3) = x + 2\).

(2) Medium Difficulty

Calculate \((3x^2 - 4x + 1) - (2x^2 + 3x - 2)\)

Change signs: Change the sign of the subtrahend to \(-2x^2 - 3x + 2\);
Convert to addition: \(3x^2 - 4x + 1 - 2x^2 - 3x + 2\);
Combine like terms: \((3x^2 - 2x^2) + (-4x - 3x) + (1 + 2) = x^2 - 7x + 3\).

(3) Hard Difficulty

Calculate \((4x^3 + 2x^2y - 5xy^2 + y^3) - (2x^3 - 3x^2y + xy^2 - 2y^3)\)

Change signs: Change the sign of the subtrahend to \(-2x^3 + 3x^2y - xy^2 + 2y^3\);
Convert to addition: \(4x^3 + 2x^2y - 5xy^2 + y^3 - 2x^3 + 3x^2y - xy^2 + 2y^3\);
Combine like terms: \((4x^3 - 2x^3) + (2x^2y + 3x^2y) + (-5xy^2 - xy^2) + (y^3 + 2y^3) = 2x^3 + 5x^2y - 6xy^2 + 3y^3\).

4. Problem-Solving Techniques

Step-by-step breakdown method: If a polynomial has many terms, first arrange the two polynomials in descending powers of a certain variable respectively. Then, perform sign transformation and mark like terms for each term one by one to avoid confusion.
Parenthesis-assisted method: For polynomial subtraction with multiple layers of parentheses (e.g., \((x^2 - (2x - 1)) - (3x^2 - x)\)), remove the parentheses and change the signs starting from the innermost parentheses. First calculate \(x^2 - 2x + 1 - 3x^2 + x\), and then combine like terms.
Substitution verification method: After completing the calculation, specific values (e.g., \(x = 2\)) can be substituted to verify whether the result is correct. For example, when calculating \((x^2 + 2x) - (x^2 - x)\), the simplified result is 3x. Substitute \(x = 2\): the left side of the original expression is \((4 + 4) - (4 - 2) = 8 - 2 = 6\), and the right side is \(3×2 = 6\), which confirms that the result is consistent.