We will learn how to subtract polynomials. The key is to “change the signs of all terms in the polynomial being subtracted, then add.” The Grade 7 rule “change subtraction into addition by changing the signs” applies here as well.
Rule for Subtracting Polynomials
To calculate (A) − (B), change the signs of all terms in B and add the result to A.
(A) − (B) ＝ A + (−B)
In other words, the key is to change the signs of all terms when removing the parentheses of the polynomial being subtracted.
Example ①
Calculate the following: (5x² + 3x − 2) − (2x² − 4x + 6)
Answer: When the parentheses being subtracted are removed, the signs are reversed: −2x² + 4x − 6.
Expression: 5x² + 3x − 2 − 2x² + 4x − 6.
x² terms: 5x² − 2x² ＝ 3x². x terms: 3x + 4x ＝ 7x. Constant: −2 − 6 ＝ −8.
Therefore, 3x² + 7x − 8.
Calculate the following: (5x² + 3x − 2) − (2x² − 4x + 6)
Answer: When the parentheses being subtracted are removed, the signs are reversed: −2x² + 4x − 6.
Expression: 5x² + 3x − 2 − 2x² + 4x − 6.
x² terms: 5x² − 2x² ＝ 3x². x terms: 3x + 4x ＝ 7x. Constant: −2 − 6 ＝ −8.
Therefore, 3x² + 7x − 8.
Vertical Subtraction
For vertical calculations, change all the signs of the polynomial being subtracted and then calculate it as addition.
Mixed Addition and Subtraction and Evaluating Expressions
For expressions combining addition and subtraction, remove the parentheses while paying attention to the sign before each pair of parentheses, then combine like terms.
Example ②
Calculate the following: (3a − 2b + 1) − (a + 4b − 3) + (2a − b)
Answer: Remove the parentheses → 3a − 2b + 1 − a − 4b + 3 + 2a − b.
a: 3a − a + 2a ＝ 4a. b: −2b − 4b − b ＝ −7b. Constant: 1 + 3 ＝ 4.
Therefore, 4a − 7b + 4.
Calculate the following: (3a − 2b + 1) − (a + 4b − 3) + (2a − b)
Answer: Remove the parentheses → 3a − 2b + 1 − a − 4b + 3 + 2a − b.
a: 3a − a + 2a ＝ 4a. b: −2b − 4b − b ＝ −7b. Constant: 1 + 3 ＝ 4.
Therefore, 4a − 7b + 4.
Key Point
- For (A) − (B), reverse all the signs of the terms in B before adding
- To avoid sign errors when removing parentheses, write out the expression before calculating
- For vertical calculations, first change the signs and then convert the subtraction to addition
- For mixed addition and subtraction, handle the signs of all terms according to the sign before the parentheses
Practice Problems
- Calculate the following: (7x + 4) − (3x − 5)
- Calculate the following: (2a² − 5a + 1) − (a² + 2a − 4)
- Calculate the following: (4x − 3y + 2) + (x + 5y − 1) − (2x − y + 6)
Answer and Explanation
- Answer:4x + 9
Explanation: Remove the parentheses → 7x + 4 − 3x + 5. x: 7x − 3x = 4x. Constant: 4 + 5 = 9. - Answer:a² − 7a + 5
Explanation: Remove the parentheses → 2a² − 5a + 1 − a² − 2a + 4. a²: 2−1=1. a: −5−2=−7. Constant: 1+4=5. - Answer:3x + 3y − 5
Explanation: Remove the parentheses → 4x − 3y + 2 + x + 5y − 1 − 2x + y − 6. x: 4+1−2=3. y: −3+5+1=3. Constant: 2−1−6=−5.