Adding & Subtracting Polynomials

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Identify like terms in polynomial expressions.
  • Add and subtract polynomials accurately.
  • Combine expressions involving variables and constants.

What Are Polynomials?

A polynomial is an expression made of terms of the form

\[ ax^n \]

where \(a\) is a coefficient and \(n\) is a non-negative integer.

Examples:

  • \(3x^2 + 7x - 5\)
  • \(x^3 - 4x\)

Key Ideas

  • Like terms have the same variable and exponent.
  • Combine like terms by adding or subtracting coefficients.

Example:
\[ 7x^2 + 3x^2 = 10x^2 \]

Common Problem Types

1. Adding Polynomials

Line up like terms and add coefficients.

2. Subtracting Polynomials

Rewrite subtraction by distributing a negative, then combine like terms.

3. Mixed Expressions

Expressions may include constants, linear terms, quadratic terms, and higher powers.

4. Simplifying Long Expressions

Combine all like terms after removing parentheses.

Strategies

  • Rewrite expressions vertically if it helps line up like terms.
  • When subtracting, distribute the negative to every term of the second polynomial.
  • Combine coefficients only when variables and exponents match exactly.
  • Keep track of signs—especially when multiple negatives appear.
  • Rewrite results in standard form: highest exponent to lowest.

Worked Examples

Example 1 — Adding Polynomials

Simplify:
\[ (5x^2 + 7x - 3) + (2x^2 - 4x + 8) \]

Solution:
Combine like terms:

\[ \begin{split} 5x^2 + 2x^2 &= 7x^2 \\ 7x - 4x &= 3x \\ -3 + 8 &= 5 \end{split} \]

Final answer:
\[ 7x^2 + 3x + 5 \]

Example 2 — Subtracting Polynomials

Subtract:
\[ (3x^3 - x + 6) - (x^3 + 4x - 2) \]

Solution:

Distribute the negative:

\[ 3x^3 - x + 6 - x^3 - 4x + 2 \]

Combine like terms:

\[ \begin{split} 3x^3 - x^3 &= 2x^3 \\ -x - 4x &= -5x \\ 6 + 2 &= 8 \end{split} \]

Final answer:
\[ 2x^3 - 5x + 8 \]

WarningCommon Mistakes
  • Adding unlike terms (e.g., \(x^2 + x \neq x^3\)).
  • Forgetting to distribute a negative when subtracting.
  • Losing track of signs when combining terms.

Practice Problems

  1. \((7x + 3) + (2x - 5)\)
  2. \((4x^2 - 6x + 1) - (x^2 + 3x - 4)\)
  3. \((8x^3 + 2x) + (x^3 - 7x)\)
  4. \((-3x^2 + 9) - (5x^2 - 1)\)
  5. \((6a - 4) + (2 - 9a)\)

1.
\[ 7x + 3 + 2x - 5 = 9x - 2 \]

2.
\[ 4x^2 - 6x + 1 - x^2 - 3x + 4 = 3x^2 - 9x + 5 \]

3.
\[ 8x^3 + 2x + x^3 - 7x = 9x^3 - 5x \]

4.
\[ -3x^2 + 9 - 5x^2 + 1 = -8x^2 + 10 \]

5.
\[ 6a - 4 + 2 - 9a = -3a - 2 \]

Summary

  • Add and subtract polynomials by combining like terms.
  • Match variables and exponents exactly before combining.
  • Distribute negatives when subtracting expressions.
  • Rewrite results in standard polynomial form.
  • Keep track of signs to avoid errors.
  • Only combine terms with the exact same \(x^n\) pattern.
  • When subtracting, turn subtraction into addition by distributing the negative.
  • Rewrite large expressions vertically to organize like terms.
  • Simplify completely before writing the final answer.