Adding and Subtracting Polynomials

1Summary 重點摘要

Learn adding and subtracting polynomials, most important is combining like terms and handling brackets.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Adding polynomialsAdding polynomials follows one principle: remove all brackets, then combine like terms. For example, (3x² + 2x) + (5x² - 3x) = 3x² + 2x + 5x² - 3x = (3+5)x² + (2-3)x = 8x² - x. Since every bracket is preceded by a plus, all signs stay unchanged, so addition is really just adding the coefficients of like terms. Work through the x² terms, then the x terms, then the constants, one class at a time; this keeps the work orderly, avoids omissions and makes the steps easy for the examiner to follow.
2. Step 2: The rule for removing bracketsThere are only two rules for removing brackets: with a plus in front, all signs inside stay as they are, so +(2x - 3) = 2x - 3; with a minus in front, every sign inside flips, so -(2x - 3) = -2x + 3. The second rule can be understood as multiplying by -1, which turns positive into negative and vice versa throughout the bracket. Since it decides success in subtraction, test yourself on a simple case such as -(x - 2) = -x + 2 before expanding a long expression.
3. Step 3: Subtracting polynomialsSubtraction is the most error-prone operation, because every term of the second bracket flips sign. For example, (3x² + 2x) - (5x² - 3x) = 3x² + 2x - 5x² + 3x, where 5x² becomes -5x² and -3x becomes +3x, and combining gives -2x² + 5x. Remember that the sign change applies to every term inside, not just the first; in the worked example, (5x² - 3x + 2) - (2x² + x - 4) expands to 5x² - 3x + 2 - 2x² - x + 4 and simplifies to 3x² - 4x + 6, with all three terms flipped and none left out.
4. Step 4: The vertical methodBesides the horizontal method, polynomials can be added or subtracted vertically: write the second expression under the first, align like terms column by column — x² terms above x² terms, constants above constants — then add or subtract row by row. The vertical layout keeps like terms aligned automatically, which is especially helpful when there are many terms; for vertical subtraction, flip the signs of the lower expression first and then add, exactly matching the horizontal rule. Whichever method you choose, mark the like terms with underlines or circles so the steps stay clear and easy to check.
5. Step 5: Substituting a valueAfter simplifying, a value of x is often substituted to produce a numerical answer, and this is done directly on the simplified expression. For example, with 8x² - x at x = 2, the value is 8(4) - 2 = 32 - 2 = 30. Simplifying before substituting has two benefits: the expression is shorter, so there is less arithmetic and less chance of error, and one simplification serves any number of substitutions. When substituting negative values, always add brackets — at x = -2, x² must be written (-2)² = 4, since missing brackets would wrongly give -4.
6. Step 6: Common errors and remediesThe most common error in this lesson is failing to flip signs after a minus, such as writing -(2x - 3) as -2x - 3 instead of the correct -2x + 3. There are two remedies: first, mark the flipped sign above every term immediately after removing the bracket and confirm before combining; second, verify by substitution — at x = 1 the original gives -(2-3) = 1 while -2(1)+3 = 1, and their agreement confirms the signs. Other slips include dropping the final constant of a bracket, and adding coefficients to powers when combining, so target these spots when reviewing.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Plus bracket signs stay same
• Minus bracket signs all flip
• Combine like terms
• Be careful with signs
• Remember brackets when substituting