Adding and Subtracting Polynomials
Overview
Adding and subtracting polynomials involves combining like terms. Like terms are terms with the same variable and the same exponent. When adding or subtracting, only the coefficients of like terms are combined.
Steps to Add or Subtract Polynomials
- Arrange the terms in each polynomial in descending order of exponents if needed.
- Combine like terms by adding or subtracting their coefficients.
- Simplify the resulting expression.
Example 1: Adding Polynomials
Consider the polynomials:
\( (3x^2 + 2x + 5) + (4x^2 - x + 7) \)
Combine like terms:
\( = (3x^2 + 4x^2) + (2x - x) + (5 + 7) \)
\( = 7x^2 + x + 12 \)
So, the sum is:
\( 7x^2 + x + 12 \)
Example 2: Subtracting Polynomials
Consider the polynomials:
\( (5x^3 - 2x + 6) - (3x^3 + x - 4) \)
Distribute the subtraction sign and combine like terms:
\( = (5x^3 - 3x^3) + (-2x - x) + (6 + 4) \)
\( = 2x^3 - 3x + 10 \)
So, the result is:
\( 2x^3 - 3x + 10 \)
Practice Questions
- Question 1: Add the following polynomials:
\( (4x^2 + 3x + 1) + (2x^2 - x + 5) \)
Solution
Combine like terms:
\( = (4x^2 + 2x^2) + (3x - x) + (1 + 5) \)
\( = 6x^2 + 2x + 6 \)
- Question 2: Subtract the following polynomials:
\( (7x^3 - x + 4) - (3x^3 + 2x - 5) \)
Solution
Distribute the subtraction sign and combine like terms:
\( = (7x^3 - 3x^3) + (-x - 2x) + (4 + 5) \)
\( = 4x^3 - 3x + 9 \)
- Question 3: Add the following polynomials:
\( (x^3 + 4x^2 - 2x) + (2x^3 - 3x^2 + x - 7) \)
Solution
Combine like terms:
\( = (x^3 + 2x^3) + (4x^2 - 3x^2) + (-2x + x) + (-7) \)
\( = 3x^3 + x^2 - x - 7 \)
- Question 4: Subtract the following polynomials:
\( (6x^2 + x - 8) - (2x^2 - 3x + 5) \)
Solution
Distribute the subtraction sign and combine like terms:
\( = (6x^2 - 2x^2) + (x + 3x) + (-8 - 5) \)
\( = 4x^2 + 4x - 13 \)
- Question 5: Add the following polynomials:
\( (3x^3 - x + 2) + (x^3 + 4x - 5) \)
Solution
Combine like terms:
\( = (3x^3 + x^3) + (-x + 4x) + (2 - 5) \)
\( = 4x^3 + 3x - 3 \)