Subtracting Polynomials: Rules, Methods, Examples
Subtracting Polynomials: Rules, Methods, Examples
What Does Subtracting Polynomials Mean?
Subtracting polynomials is the process of finding the difference between two polynomials by negating the second one and combining like terms. It is built directly on addition: once every sign in the subtracted polynomial is flipped, the operation becomes an ordinary sum.
The two rules that govern it are short:
- Flip every sign in the polynomial being subtracted: each + becomes −, each − becomes +.
- Combine like terms only, terms with the same variable and the same power, by adding their coefficients.
The result is always another polynomial whose degree does not exceed the higher of the two inputs.
Variable glossary:
| Symbol | Meaning |
|---|---|
| xxx, yyy, zzz | variables |
| like terms | terms with identical variable parts, such as 5x2 and −3x2 |
| minuend | the polynomial you subtract from |
| subtrahend | the polynomial being subtracted, whose signs flip |
How Do You Subtract Polynomials Horizontally?
The horizontal method writes both polynomials in a line, distributes the minus sign through the second parenthesis, then groups and combines like terms. It works well for short polynomials.
Compute (5x2+2x+1)−(3x2−4x+2):
Distribute the minus sign: 5x2+2x+1−3x2+4x−2.
Group like terms: (5x2−3x2)+(2x+4x)+(1−2).
Combine:
Result 2x2 + 6x - 1.
How Do You Subtract Polynomials Vertically?
The vertical method stacks the polynomials in columns by power, flips the signs of the bottom polynomial, then adds down each column. It scales well for longer polynomials and keeps the sign-flip visible.
Compute (x2−45y+35)−(7x2−12y+10):
Add down each column:
Result −6x2−33y+25.
Properties of Polynomial Subtraction
Subtraction shares one property with addition and breaks two others:
- Closure holds. The difference of two polynomials is always another polynomial.
- Commutative fails. Order matters: P−Q is not the same as Q−P.
- Associative fails. Grouping matters: (P−Q)−R is not the same as P−(Q−R).
The one safe property is closure.
Examples of Subtracting Polynomials
Example 1
Subtract (2x+1) from (5x+4).
Result 3x + 3.
Example 2
Compute (4x2+3x−5)−(x2−2x+6).
Result 3x2 + 5x - 11.
Example 3
Compute (5x2+2x+1)−(3x2−4x+2) horizontally.
Result 2x2 + 6x - 1.
Example 4
Compute (27x3−4y2+31z+9)−(16x3+32y2−8z−7).
Result 11x3−36y2+39z+16.
Example 5
Compute (3x3+4x−2)−(x3−2x2+5) vertically, inserting placeholders.
Result 2x3 + 2x2 + 4x - 7.
Example 6
Subtract (2a2−3ab+b2) from (5a2+ab−4b2).
Result 3a2 + 4ab - 5b2.
Key Takeaways
- Subtracting polynomials means flipping the sign of every term in the second polynomial, then combining like terms.
- Only like terms combine; insert zero placeholders for missing powers in the vertical method.
- Subtracting polynomials is addition with a sign change.