We will combine addition and subtraction of polynomials with multiplication and division by numbers. Finally, we will organize calculations involving fractional expressions.
Calculating Polynomials with Parentheses
Proceed in the following order:
① Remove the parentheses using the distributive property
② Combine like terms
Example ①
Calculate the following: 3(2x² + x − 4) − 2(x² − 3x + 1)
Answer: Distribute → 6x² + 3x − 12 − 2x² + 6x − 2.
x²: 6 − 2 = 4 → 4x². x: 3 + 6 = 9 → 9x. Constant: −12 − 2 = −14.
Therefore, 4x² + 9x − 14.
Calculate the following: 3(2x² + x − 4) − 2(x² − 3x + 1)
Answer: Distribute → 6x² + 3x − 12 − 2x² + 6x − 2.
x²: 6 − 2 = 4 → 4x². x: 3 + 6 = 9 → 9x. Constant: −12 − 2 = −14.
Therefore, 4x² + 9x − 14.
Polynomial Fractions
For an expression containing fractions, find a common denominator and combine into one fraction, or distribute to each term and calculate.
Example ②
Calculate the following: (3x + 2)/4 + (x − 6)/2
Answer: Make the denominators 4: (x − 6)/2 ＝ 2(x − 6)/4 ＝ (2x − 12)/4.
Adding gives: {(3x + 2) + (2x − 12)}/4 ＝ (5x − 10)/4.
Therefore, (5x − 10)/4.
Calculate the following: (3x + 2)/4 + (x − 6)/2
Answer: Make the denominators 4: (x − 6)/2 ＝ 2(x − 6)/4 ＝ (2x − 12)/4.
Adding gives: {(3x + 2) + (2x − 12)}/4 ＝ (5x − 10)/4.
Therefore, (5x − 10)/4.
Using Algebraic Expressions (Properties of Numbers)
Algebraic expressions are useful for proving properties of numbers, such as “○○ is always ○○.”
Example: “The sum of two even numbers is even”
Let two even numbers be 2m and 2n (m and n are integers) → Their sum is 2m + 2n = 2(m + n).
Since m + n is an integer, 2(m + n) is even.
Example ③
Explain why the sum of two consecutive integers is odd.
Answer: Let two consecutive integers be n and n + 1 (n is an integer).
Their sum is n + (n + 1) ＝ 2n + 1.
Since 2n + 1 has the form of an odd number, the sum of two consecutive integers is odd.
Explain why the sum of two consecutive integers is odd.
Answer: Let two consecutive integers be n and n + 1 (n is an integer).
Their sum is n + (n + 1) ＝ 2n + 1.
Since 2n + 1 has the form of an odd number, the sum of two consecutive integers is odd.
Key Point
- Calculate in the order: parentheses → distributive property → combine like terms
- For fractional expressions, finding a common denominator and combining them into one fraction makes the calculation clearer
- Even number: 2n; odd number: 2n+1; consecutive integers: n, n+1, n+2, and so on
- Organize the expression into the form of the desired conclusion and write the conclusion clearly
Practice Problems
- Calculate the following: 2(3a − b) − 3(a + 2b)
- Calculate the following: (2x + 3)/3 + (x − 4)/6
- Explain using an algebraic expression why the sum of three consecutive integers is a multiple of 3.
Answer and Explanation
- Answer:3a − 8b
Explanation: Distribute → 6a − 2b − 3a − 6b. a: 6−3=3 → 3a. b: −2−6=−8 → −8b. - Answer:(5x + 2)/6
Explanation: Make the denominators 6: (2x+3)/3 = (4x+6)/6. Adding gives (4x+6+x−4)/6 = (5x+2)/6. - Answer:Let three consecutive integers be n, n+1, and n+2. Their sum is n+(n+1)+(n+2) = 3n+3 = 3(n+1). Since n+1 is an integer, 3(n+1) is a multiple of 3.
Explanation: An expression in the form 3×(integer) is a multiple of 3. The key is factoring out the common factor 3.