Degree, terms and standard form
Some expressions are built from a variable using only adding and multiplying. How do you describe one, and which single number tells you most about it?
Where you are 10 in Polynomials
Why this exists
Cannonballs and oranges are often stacked as square pyramids: one ball on top, a 2 \times 2 layer under it, then 3 \times 3, and so on. A stack with n layers holds 1 + 4 + 9 + \dots + n^2 balls, and that sum can be written in one line:
Counting the diagonals of a polygon with n corners gives another such formula, \displaystyle \frac{n(n - 3)}{2}, met in angle sums of polygons. Both are built from a variable using nothing but multiplying and adding, so both are polynomials. Areas, volumes, counts and the paths of thrown objects keep producing expressions like these.
To work with them you need words for their parts, and one number that tells you most about each: the degree, the highest power present. Because the cannonball formula has degree three, the stack grows roughly like a cube, close to \displaystyle \frac{1}{3}n^3 balls when n is large, as Example 6 shows.
Definition
A term (also called a monomial) is a number, or a number multiplied by one or more variables, each raised to a whole-number power (0, 1, 2, 3, \dots). Examples: 7x^3, -y, 4ab^2, \displaystyle \frac{x}{2} and 5. The number in front is the term’s coefficient.
A polynomial is a term or a sum of terms. Subtracting a term counts as adding its opposite, so 5x^2 - 8x is 5x^2 + (-8x), a polynomial with two terms.
Reading coefficients takes care in three places:
- An unwritten coefficient is 1: in x^2 it is 1, and in -x^3 it is -1.
- A subtracted term has a negative coefficient: in 5x^2 - 8x the coefficient of x is -8.
- Dividing by a number is multiplying by its reciprocal: \displaystyle \frac{x}{2} = \frac{1}{2}x, coefficient \displaystyle \frac{1}{2}.
A term with no variable is the constant term. Terms with the same variables to the same powers are like terms, and combining like terms adds their coefficients.
What is not a polynomial. The variable may only be multiplied by itself a whole number of times. Each of these breaks that:
| Expression | Why it is not a polynomial |
|---|---|
| \displaystyle \frac{3}{x} | a variable in a denominator: \displaystyle \frac{3}{x} = 3x^{-1}, a negative exponent |
| x^{-2} + 1 | a negative exponent |
| \displaystyle \sqrt{x} | a variable under a root: \displaystyle \sqrt{x} = x^{1/2}, a fractional exponent |
| 2^x | the variable in the exponent |
The coefficients themselves may be any real numbers: \displaystyle \sqrt{3}\,x and \pi r^2 are polynomials, because the root and the \pi sit on numbers, not on the variable.
Degree. The degree of a term is the sum of the exponents on its variables. The term 7x^3 has degree 3; 4ab^2 = 4a^1b^2 has degree 1 + 2 = 3. A nonzero constant such as 5 has degree 0, since 5 = 5x^0 by the zero exponent rule. The degree of a polynomial is the highest degree among its terms, read after like terms are combined.
The polynomial 0 is left without a degree. It can be written 0x, 0x^2 or 0x^{50} equally well, so no power is the right one.
Names by number of terms: one term is a monomial, two a binomial, three a trinomial. With more, say “a polynomial with four terms”.
Names by degree:
| Degree | Name | Example |
|---|---|---|
| 0 | constant | -6 |
| 1 | linear | 3x + 1 |
| 2 | quadratic | x^2 - 4x |
| 3 | cubic | 2x^3 + x |
Standard form. A polynomial in one variable is in standard form when its terms run from highest degree to lowest, with the constant last. The first term is then the leading term, and its coefficient is the leading coefficient.
Here the leading coefficient is -2 and the constant term is 9.
Intuition
Look at the cannonball stack layer by layer, as in Figure 1. Each layer is a square, so layer k holds k^2 balls, and adding n squares gives a total that grows like n^3.
That is what degree measures: how many times the variable is multiplied together in the biggest term. When the variable is large, the highest power swamps the rest. For the diagonals, \displaystyle \frac{1}{2}n^2 - \frac{3}{2}n at n = 100 is 5000 - 150 = 4850: the leading term alone is already close, and the gap shrinks in proportion as n grows.
Results
Degree of a product. If p has degree m and q has degree k (both in one variable, neither zero), then pq has degree m + k, and its leading coefficient is the product of theirs. For instance (2x^3 - x)(x^2 + 4) = 2x^5 + 7x^3 - 4x has degree 3 + 2 = 5.
Degree of a sum or difference. It is at most the larger of the two degrees. It is smaller only when the leading terms cancel.
The leading term rules the far ends. For large positive or negative values of the variable, the leading term decides which way the polynomial heads, which is what you read off as end behavior.
Why it is true
Products. Multiplying every term of p by every term of q, the exponents add, as in multiplying monomials. The two leading terms give a x^m \cdot b x^k = ab\,x^{m+k}, and ab \neq 0 because neither leading coefficient is zero. Every other pair has a smaller exponent sum, so nothing can cancel the x^{m+k} term, and nothing goes higher.
Sums. Adding polynomials only adds coefficients of like terms; it never raises an exponent. So no power above the larger degree can appear. If both have the same degree and their leading coefficients add to zero, that top term disappears and the degree drops.
Worked examples
Reading the parts
List the terms of \displaystyle 8x^3 - x^2 + \frac{x}{2} - 7 with the coefficient and degree of each. Then give the degree of the polynomial, its leading coefficient and its constant term.
SolutionRead each subtraction as adding the opposite: \displaystyle 8x^3 + (-x^2) + \frac{1}{2}x + (-7).
| Term | Coefficient | Degree |
|---|---|---|
| 8x^3 | 8 | 3 |
| -x^2 | -1 | 2 |
| \displaystyle \frac{x}{2} | \displaystyle \frac{1}{2} | 1 |
| -7 | -7 | 0 |
The coefficient of -x^2 is the unwritten 1 with the minus sign, so -1. The x in \displaystyle \frac{x}{2} has exponent 1, and the constant has degree 0.
The highest term degree is 3, so the polynomial is cubic, with four terms. It is already in standard form, so the leading coefficient is 8. The constant term is -7.
Polynomial or not
Which of these are polynomials? (a) \displaystyle \frac{4}{x} + 1 (b) \displaystyle \frac{x^2 - 3}{5} (c) \displaystyle \sqrt{3}\,y + 2 (d) \displaystyle 3\sqrt{x} - x (e) 6t^{-2}
Solution(a) Not a polynomial. \displaystyle \frac{4}{x} = 4x^{-1}, a negative power.
(b) A polynomial. The division is by a number, so it is \displaystyle \frac{1}{5}x^2 - \frac{3}{5}, a quadratic binomial.
(c) A polynomial. The root is on the number 3, so \displaystyle \sqrt{3} is just a coefficient. Degree 1.
(d) Not a polynomial. \displaystyle \sqrt{x} = x^{1/2}, and \displaystyle \frac{1}{2} is not a whole number.
(e) Not a polynomial. The exponent -2 is negative; it is \displaystyle \frac{6}{t^2} in disguise.
Several variables
Find the degree of each term of 2x^3y^2 - 9x^4 + xy - 5, and the degree of the polynomial.
SolutionAdd the exponents in each term:
| Term | Exponents | Degree |
|---|---|---|
| 2x^3y^2 | 3 + 2 | 5 |
| -9x^4 | 4 | 4 |
| xy | 1 + 1 | 2 |
| -5 | none | 0 |
The polynomial has degree 5. The largest single exponent is the 4 on x^4, but degree counts all the variables in a term together, and x^3y^2 multiplies five variables.
Standard form and names
Write each in standard form. Give the leading coefficient and the constant term, and name it by degree and by number of terms. (a) 9 + 4x^3 - x (b) 3x - 5x^2 (c) -12
Solution(a) Ordering by degree, 3, then 1, then 0, gives 4x^3 - x + 9. There is no x^2 term; its coefficient is 0 and it is simply left out. The leading coefficient is 4 and the constant term is 9: a cubic trinomial.
(b) -5x^2 + 3x. The leading coefficient carries its sign, so it is -5. There is no constant, so the constant term is 0: a quadratic binomial.
(c) -12 is one term of degree 0: a constant monomial. Its leading coefficient and its constant term are both -12.
Simplify before you read the degree
Simplify, then give the degree. (a) (x^2 + 3x) - (x^2 - 1) (b) (5y^3 - 2y + 8) - (5y^3 - 2y) (c) (x^2 + 3x) - (3x + x^2)
SolutionSubtracting a bracket subtracts every term in it, so each sign inside flips. Combining like terms is adding and subtracting polynomials; here only the result matters.
(a) x^2 + 3x - x^2 + 1 = 3x + 1. Degree 1, a linear binomial. The x^2 terms cancel, so the degree dropped from the 2 it seemed to have.
(b) 5y^3 - 2y + 8 - 5y^3 + 2y = 8. Degree 0.
(c) x^2 + 3x - 3x - x^2 = 0, the zero polynomial, which has no degree.
Two counts as polynomials
Araminta counts cannonballs in a square stack, \displaystyle C = \frac{2n^3 + 3n^2 + n}{6} for n layers. (a) Write C in standard form and give its degree and leading coefficient. Check it for n = 4. (b) Compare C with its leading term for n = 30. (c) Write the diagonal count \displaystyle \frac{n(n - 3)}{2} as a polynomial in standard form.
Solution(a) Dividing each term by 6:
A cubic trinomial with leading coefficient \displaystyle \frac{1}{3} and constant term 0, which makes sense: no layers, no balls. For n = 4: \displaystyle \frac{2(64) + 3(16) + 4}{6} = \frac{180}{6} = 30, matching 1 + 4 + 9 + 16 from Figure 1.
(b) \displaystyle C = \frac{2(27000) + 3(900) + 30}{6} = \frac{56730}{6} = 9455, while \displaystyle \frac{1}{3}(30)^3 = 9000. The leading term alone gives most of the count. A 30 \times 30 \times 30 cube of balls would hold 27000, so the stack fills \displaystyle \frac{9455}{27000}, about 0.35 of the cube: close to a third, and closer as the stack grows, since the leading term \displaystyle \frac{1}{3}n^3 takes a larger share of the total. A solid pyramid fills exactly a third of its box, as in the volume of a pyramid, and a stack of balls is a lumpy version of one.
(c) Distribute, then divide each term by 2:
A quadratic binomial, leading coefficient \displaystyle \frac{1}{2}, constant term 0. For a hexagon, \displaystyle \frac{1}{2}(36) - \frac{3}{2}(6) = 18 - 9 = 9.
Boundaries
Whole-number exponents only. A variable in a denominator, under a root, or with a negative or fractional exponent makes an expression something other than a polynomial, and it has no degree. \displaystyle \frac{x^2 + x}{x} is not a polynomial as written, even though it matches x + 1 at every x except 0.
Degree is read after simplifying. (x^2 + 3x) - (x^2 - 1) looks quadratic and is linear.
The zero polynomial has no degree. Every other constant has degree 0.
Standard form and leading coefficient are for one variable. With several variables the degree is still defined, but “highest first” can be ambiguous: x^2y + xy^2 has two terms of degree 3.
The product rule needs both factors nonzero. Multiplying by the zero polynomial gives 0, which has no degree to add.
Where this goes next
With the vocabulary in place, polynomials can be combined. Adding and subtracting polynomials is combining like terms across two of them, and multiplying polynomials distributes term by term, where the degree of the answer can be predicted before you start. Degree two gets its own study in quadratic expressions and their graphs, parabolas. Running the other way, factoring trinomials splits a quadratic, when it can, into two linear factors, 1 + 1 = 2, and polynomial long division lowers the degree one step at a time.
Exercises
Polynomial or not
Decide whether each is a polynomial. If it is, give its degree.
- \displaystyle 7x^2 - \frac{x}{3}
- \displaystyle \frac{3}{y^2} + y
- \displaystyle 5 - \sqrt{t}
- \pi r^2
Terms, coefficients and degrees
- List the terms of -x^5 + 6x^2 - x + 10 with their coefficients. Give the constant term, the degree and the leading coefficient.
- Find the degree of each term of 4m^2n^3 - m^4 + 3mn and the degree of the polynomial.
Standard form and names
Write in standard form, give the leading coefficient and the constant term, and name the polynomial by degree and by number of terms.
- 3 - 2x^2 + 5x^4
- x - x^3
- \displaystyle \frac{1}{2} + 6y^2 - y
- Write a linear binomial in x with leading coefficient -4 and constant term 9. Is there only one?
Predicting the degree
- Without adding, give the degree and leading coefficient of (2x^4 - x + 1) + (x^3 + 6x^2), and say why the degree cannot drop.
- Simplify (x + 3)(x + 4) - x(x + 7) and give its degree.
- Without multiplying out, give the degree and leading coefficient of (3x^2 - 1)(2x^3 + x). Then multiply out to check.
Using polynomials
- A storage box is x dm tall, x + 2 dm wide and 2x - 1 dm long, so its volume is x(x + 2)(2x - 1) cubic decimeters. Predict the degree and leading coefficient, then write the volume in standard form and check it for x = 3.
- Use \displaystyle \frac{1}{2}n^2 - \frac{3}{2}n to count the diagonals of a polygon with 20 sides. What does the leading term alone give?
Reasoning
- Explain why the sum of two cubic polynomials can have degree less than 3 but never more than 3. Give a pair whose sum has degree 2.
- Bertil says 0 should have degree 0, just like 5. Explain why it is left without a degree.
- Berenice says \displaystyle \frac{x^2 + x}{x} is not a polynomial. Balthazar says it equals x + 1, so it is. Who is right?
Answers
- A polynomial of degree 2: \displaystyle 7x^2 - \frac{1}{3}x, a quadratic binomial.
- Not a polynomial. \displaystyle \frac{3}{y^2} = 3y^{-2}, a negative exponent.
- Not a polynomial. \displaystyle \sqrt{t} = t^{1/2}, a fractional exponent.
- A polynomial of degree 2: a monomial in r with coefficient \pi, a number.
- Terms -x^5, 6x^2, -x, 10 with coefficients -1, 6, -1, 10. Constant term 10, degree 5, leading coefficient -1.
- 4m^2n^3 has degree 2 + 3 = 5, -m^4 has degree 4, 3mn has degree 1 + 1 = 2. The polynomial has degree 5.
- 5x^4 - 2x^2 + 3. Leading coefficient 5, constant term 3: a trinomial of degree 4.
- -x^3 + x. Leading coefficient -1, constant term 0: a cubic binomial.
- \displaystyle 6y^2 - y + \frac{1}{2}. Leading coefficient 6, constant term \displaystyle \frac{1}{2}: a quadratic trinomial.
- -4x + 9, and it is the only one. Linear means the terms are an x term and a constant, and both coefficients are fixed.
- Degree 4, leading coefficient 2. Only the first polynomial has an x^4 term, so nothing can cancel 2x^4. In full the sum is 2x^4 + x^3 + 6x^2 - x + 1.
- (x^2 + 7x + 12) - (x^2 + 7x) = 12. Degree 0.
- Degree 2 + 3 = 5, leading coefficient 3 \cdot 2 = 6. Check: 6x^5 + 3x^3 - 2x^3 - x = 6x^5 + x^3 - x.
- Degree 1 + 1 + 1 = 3, leading coefficient 1 \cdot 1 \cdot 2 = 2. Multiplying, (x + 2)(2x - 1) = 2x^2 + 3x - 2, so the volume is 2x^3 + 3x^2 - 2x. For x = 3: 54 + 27 - 6 = 75 = 3 \cdot 5 \cdot 5.
- \displaystyle \frac{1}{2}(400) - \frac{3}{2}(20) = 200 - 30 = 170 diagonals. The leading term alone gives 200.
- Adding only combines like terms, which never raises an exponent, so no power above 3 can appear. The x^3 terms cancel when their coefficients are opposites: (4x^3 + x^2) + (-4x^3 + 3x - 1) = x^2 + 3x - 1, degree 2.
- 5 = 5x^0, and no other way of writing it with a nonzero coefficient uses another power, so 0 is the right degree. But 0 = 0x = 0x^2 = 0x^7: every power works equally well, so no single degree is the right one.
- Berenice. As written it has the variable in a denominator. It agrees with x + 1 at every x except 0, where it has no value at all while x + 1 equals 1. They are different expressions.