Polynomial operations become reliable when terms are organized by descending degree. Addition and subtraction combine like terms; multiplication distributes factors; division reverses multiplication while tracking any remainder.
Learning objectives
- Identify terms, coefficients, variables, constants, and degree.
- Classify monomials, binomials, trinomials, and general polynomials.
- Add and subtract by aligning like terms, including zero placeholders.
- Multiply or divide every term by a monomial.
- Use polynomial long division and verify \(P(x)=D(x)Q(x)+R(x)\).
Polynomial vocabulary and degree
Anatomy of a polynomial
The labels identify terms, coefficients, variables, exponents, the constant, and the degree without relying on color alone.
- Leading term
- \(6x^4\)
- Coefficient 6, variable x, exponent 4
- Term
- \(-3x^2\)
- Coefficient -3 and exponent 2
- Linear term
- \(8x\)
- Exponent 1 is understood
- Constant
- \(-11\)
- No variable; degree 0
Polynomial degree: 4
| Feature | Meaning | Example |
|---|---|---|
| Term | A coefficient-variable product separated by addition or subtraction | \(-4x^3\) |
| Coefficient | The numerical factor of a term | \(-4\) in \(-4x^3\) |
| Constant | A term with no variable; its degree is \(0\) | \(9\) |
| Degree of a term | Sum of variable exponents | \(3x^2y^4\) has degree \(6\) |
| Degree of a polynomial | Greatest term degree after simplification | \(5x^4-x+2\) has degree \(4\) |
Adding and subtracting polynomials
A sign-safe method
- Remove grouping
For subtraction, distribute the negative sign to every term in the second polynomial.
- Align powers
Write terms in descending degree and insert \(0x^k\) placeholders when useful.
- Combine
Add coefficients only for identical variable parts.
- Check
Evaluate the original and simplified expressions at a safe value such as \(x=2\).
Subtract without losing signs
Simplify \((4x^3-2x+7)-(x^3+5x^2-6x-3)\).
- Distribute subtraction
Write \(4x^3-2x+7-x^3-5x^2+6x+3\).
- Combine like terms
The cubic, quadratic, linear, and constant coefficients become \(3,-5,4,10\).
Multiplying and dividing by a monomial
Distribute in both directions
Multiply
Multiply every coefficient and add exponents of matching bases: \(-3x^2(2x^3-x+4)=-6x^5+3x^3-12x^2\).
Divide
Divide every term separately and subtract exponents: \(\frac{18x^5-12x^3}{6x^2}=3x^3-2x\).
Polynomial long division
Polynomial long division with a missing term
Polynomial long division showing each quotient, product, subtraction, and remainder step.
Check a division identity
If \(P(x)=(x+3)(2x^2-x+4)-5\), what are the divisor, quotient, and remainder?
- \(x+3, 2x^2-x+4, -5\)
- \(x+3, 2x^2-x-1, 5\)
- \(2x^2-x+4, x+3, 5\)
Show answer and explanation
Answer: Divisor \(x+3\), quotient \(2x^2-x+4\), remainder \(-5\).
Match the expression directly to \(P=DQ+R\); the final signed constant is the remainder.
Key takeaways
What to remember
- Combine only like terms.
- Distribute a subtraction sign to every term.
- Use zero placeholders for missing powers.
- Verify long division using \(P=DQ+R\).
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.