We will learn how to add polynomials. The basic idea is to combine like terms. Think of this as the linear-expression calculations from Grade 7, with only the number of types of terms increased.
To add polynomials, simply remove the parentheses and combine like terms.
The parentheses in (A) + (B) can be removed as they are (the signs remain unchanged).
Example ①
Calculate the following: (3x² + 2x − 5) + (x² − 4x + 7)
Answer: Remove the parentheses → 3x² + 2x − 5 + x² − 4x + 7.
x² terms: 3x² + x² ＝ 4x². x terms: 2x − 4x ＝ −2x. Constants: −5 + 7 ＝ 2.
Therefore, 4x² − 2x + 2.
Calculate the following: (3x² + 2x − 5) + (x² − 4x + 7)
Answer: Remove the parentheses → 3x² + 2x − 5 + x² − 4x + 7.
x² terms: 3x² + x² ＝ 4x². x terms: 2x − 4x ＝ −2x. Constants: −5 + 7 ＝ 2.
Therefore, 4x² − 2x + 2.
Vertical Calculation (Aligning Like Terms Vertically)
When there are many terms, aligning like terms vertically and calculating as in column arithmetic helps prevent mistakes.
3x² + 2x − 5 ＋) x² − 4x + 7 ───────────────── 4x² − 2x + 2
Adding Three or More Polynomials
With three or more polynomials, you can calculate by removing all parentheses and combining like terms. You may change the order (commutative property) or calculate after grouping terms (associative property).
Example ②
Calculate the following: (2a − 3b) + (−a + 5b) + (4a + b)
Answer: Remove the parentheses → 2a − 3b − a + 5b + 4a + b.
a terms: 2a − a + 4a ＝ 5a. b terms: −3b + 5b + b ＝ 3b.
Therefore, 5a + 3b.
Calculate the following: (2a − 3b) + (−a + 5b) + (4a + b)
Answer: Remove the parentheses → 2a − 3b − a + 5b + 4a + b.
a terms: 2a − a + 4a ＝ 5a. b terms: −3b + 5b + b ＝ 3b.
Therefore, 5a + 3b.
Key Point
- The parentheses in (A) + (B) can be removed as they are (the signs inside do not change)
- Find like terms and add their coefficients
- When there are many terms, arranging them vertically reduces mistakes
- Terms are not like terms unless their variable types and exponents match
Practice Problems
- Calculate the following: (4x − 7y) + (−2x + 3y)
- Calculate the following: (a² + 2a − 3) + (3a² − 5a + 8)
- Calculate the following: (5x + 2y − 1) + (−3x + y + 4) + (x − 4y + 2)
Answer and Explanation
- Answer:2x − 4y
Explanation: x terms: 4x − 2x = 2x. y terms: −7y + 3y = −4y. Therefore, 2x − 4y. - Answer:4a² − 3a + 5
Explanation: a² terms: 1+3=4 → 4a². a terms: 2−5=−3 → −3a. Constant: −3+8=5. - Answer:3x − y + 5
Explanation: x: 5−3+1=3 → 3x. y: 2+1−4=−1 → −y. Constant: −1+4+2=5.