Polynomial Operations

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Skill: polynomials

Polynomial Operations

Meet the polynomial: a sum of power-terms. Learn to name one by its degree and its terms, write it in standard form, evaluate it, and then add, subtract, and multiply polynomials without losing a single term.

1 Understand

The core idea in plain language.

What it is. A polynomial is a sum of terms. Each term is a number (the coefficient) times a variable raised to a whole-number power, like 3x2. The degree is the highest exponent. The number of terms gives it a family name: 1 term is a monomial, 2 a binomial, 3 a trinomial. Standard form lists terms from highest degree down.

Why it matters. Polynomials are the vocabulary of algebra. Factoring, quadratic equations, and graphing all assume you can read a polynomial the way you read a sentence — degree first, then terms.

Where it is used. Area and volume formulas with variables · projectile paths (quadratic functions) · computer animation curves.

2 See It

Diagrams that make the idea visual.

Anatomy of a polynomial

Read 4x3 − 2x + 7 left to right: the highest exponent is 3, so the degree is 3. There are three terms, so it is a trinomial. It is already in standard form — exponents falling 3, 1, 0.

4x3 − 2x + 7 leading term 4x³ degree = 3 3 terms → trinomial standard form: exponents fall 3 → 1 → 0
Anatomy of 4x³ − 2x + 7: degree 3, three terms (trinomial), in standard form.

Only identical twins combine

Like terms share the same variable and the same exponent. 3x2 and −x2 merge into 2x2; 2x cannot join them — different exponent, different family.

3x² −x² 2x +4 these two merge → 2x² 2x and +4 have no partners — they stay
Like terms share variable AND exponent. 3x² and −x² merge; 2x and +4 have no partners.

3 Worked Examples

Follow each step. The pattern is always the same.

Example 1 Classify: 4x³ − 2x + 7
  1. Highest exponent is 3 → degree 3.
  2. Count the terms: 4x3, −2x, 7 → 3 terms → trinomial.
4x³ − 2x + 7 is a degree-3 trinomial.

Check: Exponents present: 3, 1, 0. Highest is 3; nothing else to combine.

Example 2 Add: (3x² + 2x − 5) + (x² − 7x + 4)
  1. Group like terms: (3x2 + x2) + (2x − 7x) + (−5 + 4).
  2. Combine each group: 4x2, −5x, −1.
(3x² + 2x − 5) + (x² − 7x + 4) = 4x² − 5x − 1

Check: No like terms remain uncombined; degree stayed 2.

Example 3 Subtract: (5x³ − 2x) − (2x³ + x² − 9)
  1. Distribute the minus to every term inside: 5x3 − 2x − 2x3 − x2 + 9.
  2. Group and combine: (5x3 − 2x3) − x2 − 2x + 9.
(5x³ − 2x) − (2x³ + x² − 9) = 3x³ − x² − 2x + 9

Check: The −9 became +9 and +x² became −x² — the minus hit everything.

Example 4 Evaluate: 2x² − 3x + 1 at x = 2
  1. Substitute: 2(2)2 − 3(2) + 1.
  2. Compute: 8 − 6 + 1 = 3.
At x = 2, 2x² − 3x + 1 = 3.

Check: Exponents first (2² = 4), then multiply, then add.

Example 5 Multiply: (x + 4)(x − 6)
  1. FOIL — First, Outer, Inner, Last.
  2. x2 − 6x + 4x − 24, then combine the middle terms.
(x + 4)(x − 6) = x² − 2x − 24

Check: Substitute x = 0: (4)(−6) = −24 and 0 − 0 − 24 = −24.

4 Common Mistakes

These errors show up on almost every quiz. Spot them now and they will never cost you points.

Mistake 1: x² + x² = x⁴
Wrong
Adding the exponents: x2 + x2 = x4.
Right
2x2 — adding combines like terms; exponents change only when multiplying.
Rule: add coefficients, keep the exponent. x² + x² = 2x².
Mistake 2: the minus only hits the first term
Wrong
5x2 − (2x2 − 3x) = 3x2 − 3x.
Right
3x2 + 3x — the minus flips every sign inside: −(−3x) = +3x.
Rule: subtracting a polynomial flips every sign inside the parentheses.
Mistake 3: degree is the first exponent you see
Wrong
The degree of 7x2 − x5 + 3 is 2.
Right
Degree 5 — degree is the highest exponent, wherever it sits.
Rule: scan every term; the biggest exponent wins.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Classify −3x⁴ + x² − 8: what are its degree and number of terms?
Answer
Degree 4, 3 terms (trinomial).
2. Add: (2x² − x + 3) + (x² + 4x − 5).
Answer
3x² + 3x − 2.
3. Evaluate 5x³ − 2x at x = −1.
Answer
5(−1)³ − 2(−1) = −5 + 2 = −3.

Key Points to Remember

  • A polynomial is a sum of terms; each term is coefficient × variablewhole-number power.
  • Degree = highest exponent. Terms: 1 → monomial, 2 → binomial, 3 → trinomial.
  • Standard form lists terms from highest degree down.
  • Add/subtract: combine like terms (same variable, same exponent).
  • Subtracting flips every sign inside the parentheses.
  • Multiply: distribute every term to every term (FOIL for two binomials).
  • Evaluate by substituting, then exponents → multiply → add.
Practice problems Try the tool See it on FiboMap
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