Subtracting Polynomials: Mastering the Negative Sign

2. Distribute the Negative

Remove the parentheses by distributing the negative sign to every term inside the second polynomial. This is the most vital step. If a term was $+3x^2$, it becomes $-3x^2$. If a term was $-5x$, it becomes $+5x$.

3. Combine Like Terms

Once the parentheses are gone and the signs are updated, you are left with a long string of terms. Group the "like terms"—terms that have the exact same variable and exponent—and add or subtract their coefficients.

Worked Example

Let's put these steps into practice with a concrete example. Suppose we want to solve the following:

$(5x^2 + 3x - 8) - (2x^2 - 4x + 2)$

Step 1: Rewrite the expression The expression is already set up with parentheses, which helps us see exactly what needs to be distributed.

Step 2: Distribute the negative We apply the negative sign to $2x^2$, $-4x$, and $+2$.

$$\begin{aligned} (5x^2 + 3x - 8) - 2x^2 + 4x - 2 \end{aligned}$$

Notice how $-4x$ became $+4x$ and $+2$ became $-2$.

Step 3: Combine like terms Now, we group the terms with $x^2$, the terms with $x$, and the constant terms:

$$\begin{aligned} (5x^2 - 2x^2) + (3x + 4x) + (-8 - 2) \end{aligned}$$

Performing the arithmetic gives us our final answer:

$$\begin{aligned} 3x^2 + 7x - 10 \end{aligned}$$

Common Pitfalls to Avoid

Even students who understand the concept can fall into these traps:

• The "First Term Only" Error: This occurs when a student subtracts the first term but leaves the rest of the polynomial unchanged. Always double-check that every term in the second polynomial has had its sign flipped.
• Sign Confusion with Negatives: Subtracting a negative is the same as adding a positive. If you see $- (-5x)$, immediately rewrite it as $+ 5x$ to avoid mental fatigue.
• Misidentifying Like Terms: Remember that $x^2$ and $x$ are not like terms. You cannot combine them. Only terms with the same variable and the same exponent can be merged.

Exam Tip: After you finish a subtraction problem, do a quick "sign check." Look at the original second polynomial and verify that every single term in your expanded version has the opposite sign.

By mastering the distribution of the negative sign, you turn a complex-looking subtraction problem into a simple addition and grouping exercise. Practice these steps consistently to build the muscle memory needed for more advanced algebra.