Polynomials and Factoring
Polynomials: Terminology, Degree and Standard Form
Monomials, binomials and trinomials, how the degree of a multi-variable term is counted, what makes two terms alike, and why standard form is worth the trouble.
What this page covers
- Name a polynomial by its number of terms and state its degree
- Count the degree of a term in several variables correctly
- Decide whether two terms are like, and collect them
- Add and subtract polynomials without losing a sign
What counts as a polynomial
A polynomial is a sum of terms, each of which is a number multiplied by variables raised to whole-number powers. The restriction on the powers is the whole of the definition, and it is what makes polynomials well behaved: they can be added, subtracted and multiplied without ever leaving the family, and they are defined and smooth for every value of their variables.
| Expression | Polynomial? | Reason |
|---|---|---|
| 3x2 - 7x + 1 | Yes | Whole-number powers throughout |
| x3y + 3pi - 2x | Yes | Several variables is fine; π is a constant coefficient |
| 1x + 3 | No | 1x = x-1, a negative power |
| √x + 1 | No | √x = x1/2, a fractional power |
| 2x | No | The variable is in the exponent, not the base |
| 7 | Yes | A constant is 7x0, a polynomial of degree 0 |
Naming by number of terms
The first three counts have names that are used constantly, so they are worth fixing early.
| Name | Terms | Source example |
|---|---|---|
| Monomial | One | x2 |
| Binomial | Two | x + 5 |
| Trinomial | Three | x2 + 5x + 6 |
| Polynomial | Any number | x3y + 3pi - 2x |
In the binomial x + 5 the source notes label the two parts explicitly: x is the variable term and 5 is the constant term. Beyond three terms nobody bothers with a special name, and 'polynomial' is used throughout.
- Term
- One product of a coefficient and powers of variables
- Coefficient
- The numerical factor of a term; in -7x it is -7
- Constant term
- The term with no variable; equivalently the coefficient of x0
- Leading coefficient
- The coefficient of the highest-degree term
Degree: the highest combined power
In one variable the degree is simply the largest exponent. With several variables the rule needs care, and the source notes state it precisely.
The degree of a term is the sum of the exponents of all its variables. The degree of a polynomial is the largest degree among its terms — in the wording of the notes, the highest combined power in any term.
Worked example — the source's own case
Find the degree of x3 + xy4 - 5.
| Term | Exponents | Degree of term |
|---|---|---|
| x3 | 3 on x | 3 |
| xy4 | 1 on x, 4 on y | 1 + 4 = 5 |
| -5 | none | 0 |
The largest is 5, so this is a polynomial of degree 5. Source example, Week 1, page 6.
The word combined is doing real work. It is easy to glance at xy4, see the visible exponent 4, and call the polynomial degree 4. The invisible exponent 1 on x counts, and the term has degree 5.
Degrees up to three have their own names, which recur throughout the rest of the subject.
| Degree | Name | General form | Source example |
|---|---|---|---|
| 0 | Constant | y = a0 | y = 5 |
| 1 | Linear | y = a1x + a0 | y = 2x - 7 |
| 2 | Quadratic | y = a2x2 + a1x + a0 | y = 2x2 + 3x + 5 |
| 3 | Cubic | y = a3x3 + a2x2 + a1x + a0 | y = -2x3 + 5x2 - 2x + 7 |
The source works through y = 2x4 - 3x3 + x and reads off the coefficients as a4 = 2, a3 = -3, a2 = 0, a1 = 1, a0 = 0. Recording the zero coefficients matters: long division and synthetic methods both need a placeholder wherever a power is missing.
Like terms
Two terms are like when they carry the same variables raised to the same powers. The coefficients play no part, and neither does the order in which the variables happen to be written.
The source gives 3x2y3 and π y3x2 as like terms. Rearranged, the second is π x2y3; both carry x2y3, so they combine to (3 + π)x2y3. The commutative law for multiplication is what licenses the rearrangement.
| Pair | Like? | Reason |
|---|---|---|
| 3x2y3 and π y3x2 | Yes | Same variables, same powers, written in a different order |
| 5x2 and 5x3 | No | Different powers of x |
| 4xy and 4x | No | The second has no y |
| -2ab2 and 7b2a | Yes | Both carry ab2 |
| 3 and -8 | Yes | Both are x0 terms |
Collecting like terms is the only simplification available to a sum of polynomials. Unlike terms simply stay where they are; x2 + x does not reduce further, however much it may look as though it should.
Standard form
A polynomial is in standard form when its terms are written in descending order of degree. The value is unaffected — addition commutes — but three practical benefits follow.
Comparison
Two polynomials in standard form can be compared term by term without hunting for matching powers.
Leading term
The degree and the leading coefficient are both visible at a glance, and together they govern the shape of the graph far from the origin.
Division
Long division assumes descending order and a placeholder for every missing power. Out of order, the algorithm cannot start.
Adding and subtracting
The whole procedure is two steps, and the source states them in exactly this order.
Worked example — addition
A plus sign before a bracket changes nothing inside it, so the brackets can simply be dropped.
Substitute x = 1. The two originals give (1 - 3 + 8 + 7) + (-5 - 12 + 3) = 13 - 14 = -1. The answer gives -4 - 3 - 4 + 10 = -1. Agreement.
Worked example — subtraction
Every one of the three terms inside the bracket changes sign. The y terms cancel exactly, which is why the answer has only two terms.
At y = 2: originals give (4 - 10 + 7) - (12 - 10 + 12) = 1 - 14 = -13. Answer gives -8 - 5 = -13. Agreement.
Distributing a leading minus over only the first term. Writing -(3y2 - 5y + 12) = -3y2 - 5y + 12 changes two signs out of three and is wrong. The minus multiplies the whole bracket: -3y2 + 5y - 12. Where a subtraction is at all involved, write the expanded bracket on its own line before collecting anything.
Degree under the operations
Knowing what the degree does under each operation is a cheap and effective check on a long calculation.
| Operation | Resulting degree | Caveat |
|---|---|---|
| Addition or subtraction | At most the larger of the two degrees | Can be less if the leading terms cancel |
| Multiplication | Exactly the sum of the two degrees | Holds over the reals, where the leading coefficients cannot multiply to zero |
| Division with remainder | Quotient has degree ° f - ° g | The remainder has degree strictly below ° g |
The cancellation caveat is real. Adding x3 + 1 and -x3 + x gives x + 1, of degree 1, not 3. In the worked example above the x3 terms did not cancel, so the degree stayed at 3.
Multiplication of degrees is used constantly without being noticed. It is the reason a quadratic times a quadratic gives a quartic, and the reason the quotient in Polynomial Long Division has the degree it does.
Common mistakes
| Mistake | Correct | Why it happens |
|---|---|---|
| °(xy4) = 4 | °(xy4) = 5 | The invisible exponent 1 on x was not counted |
| -(a - b) = -a - b | -(a - b) = -a + b | The minus was distributed over one term only |
| x2 + x = x3 | No simplification | Adding terms was confused with multiplying them |
| 3x2y3 and 3x3y2 treated as like | They are unlike | The powers were matched loosely |
| Omitting a missing power before dividing | Write x3 + 0x2 + 0x - 2 | The gap silently misaligns every subsequent column |
| Calling √x + 1 a polynomial | It is not | The fractional power x1/2 was overlooked |
Frequently asked questions
Is 1x + 3 a polynomial?
No. 1x is x-1, and a polynomial admits only non-negative whole-number powers of its variables. The same rules out √x, which is x1/2. Expressions of that kind are rational or algebraic, not polynomial.
What is the degree of a constant?
Zero, because 5 = 5x0. The single exception is the constant 0, whose degree is normally left undefined: no choice of value makes the rule °(fg) = ° f + ° g work for it.
Does the order of terms matter?
Not to the value, since addition commutes. It matters to the reader. Descending order of degree puts the leading coefficient first, makes two polynomials easy to compare, and is what every division algorithm assumes.
Are 3x2y3 and π y3x2 like terms?
Yes. Like terms need the same variables raised to the same powers; the coefficients are irrelevant and so is the order in which the variables are written. Here both terms carry x2y3, so they combine to (3 + π)x2y3.
Source. Handwritten teaching notes, Week 1, pages 6-7.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.