Math101 · Independent Ontario learning library
Math101learn.math101.caSubtracting Polynomials
Subtracting a polynomial means adding its additive inverse. The negative sign before a bracket changes every term inside; after distribution, like terms combine by adding signed coefficients.
Polynomial subtraction compares models and is needed in division, finite differences, function operations, and algebraic simplification.
Intuition and core definition
Subtracting a polynomial means adding its additive inverse. The negative sign before a bracket changes every term inside; after distribution, like terms combine by adding signed coefficients. Polynomial subtraction is not commutative, so order matters.
Notation, language, and conditions
$P(x)-Q(x)=P(x)+[-Q(x)]$. In vertical layout, align equal powers and subtract each coefficient. Missing powers have coefficient zero. Parentheses are essential around the entire subtracted polynomial.
Why this idea matters
Subtracting a polynomial changes the sign of every term in the subtracted group before like terms may be combined.
A dependable method
- Write both polynomials in standard form with missing-power placeholders if useful.
- Distribute the subtraction sign to every term of the second polynomial.
- Remove parentheses and group like powers.
- Add the signed coefficients.
- Check by adding the result to the subtracted polynomial and recovering the first.
Worked example
Representations and interpretation
Coefficient vectors turn subtraction into componentwise signed subtraction: $(5,-2,0,4)-(3,1,-6,7)=(2,-3,6,-3)$. Algebra tiles model the additive inverse by flipping every tile’s sign.
Reasoning about variations
The degree may fall through cancellation. Subtracting a polynomial from itself gives the zero polynomial, whose degree is not treated like an ordinary finite degree.
Common mistakes
How to check your work
- Add the difference and the second polynomial to recover the first.
- Evaluate original and simplified expressions at $x=0$ and $x=1$.
- Audit that every term in the second bracket changed sign exactly once.
Practice
- Simplify $(4x^2+3x-1)-(x^2-5x+6)$.
- Compute $(2a-b)-(-a+3b)$.
- Is $P-Q$ always equal to $Q-P$?
Answers and brief solutions
Show answers
- $3x^2+8x-7$ Negate the entire second polynomial, then combine like terms.
- $3a-4b$ The second bracket becomes $a-3b$.
- No Reversing order gives the additive inverse unless the polynomials are equal.
Synthesis and transfer
Comparing two polynomial cost models requires distributing the subtraction across all terms; adding the difference back to the second model should reconstruct the first.
If planned cost is $P(x)=4x^2+7x+20$ and actual cost is $A(x)=3x^2+9x+18$, then $P-A$ is $4x^2+7x+20-(3x^2+9x+18)=x^2-2x+2$. Parentheses ensure the minus changes all three signs in the second polynomial. Adding this difference back to $A$ reconstructs $P$, giving a strong structural check. Reversing the comparison produces the negative polynomial, so contextual wording determines order. At a chosen input, numerical evaluation of both original models and their symbolic difference should agree, helping distinguish a sign-distribution error from later collection of like terms.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Simplify $(4x^2+3x-1)-(x^2-5x+6)$.
- Negate the entire second polynomial, then combine like terms.
End of lesson
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