Terms, coefficients and constants
In 5x^2 - 3x + 7 the 5, the 3 and the 7 do three different jobs. How do you point at one piece of an expression without rewriting the whole thing?
Where you are 5 in Variables
Why this exists
Even a short expression has parts doing different jobs. In 5x^2 - 3x + 7 there are three numbers. The 5 multiplies x^2. The 3 multiplies x, and a minus sign comes with it. The 7 is just 7, whatever number x stands for. To say anything about one of those pieces (“double the second one”, “these two can be put together”, “this part is the fixed charge”) you need a name for it, or you end up rewriting the whole expression every time you want to point.
The names also carry meaning. Suppose a phone plan costs C = 8g + 20 dollars a month when you use g gigabytes. The 8 is the price of each gigabyte, and the 20 is what you pay even if you use nothing at all. Calling the 8 a coefficient and the 20 a constant says exactly that: one part grows with g, and the other part stays put. That split, a part that changes with the letter and a part that does not, sits inside almost every formula you will meet.
Definition
An expression is built by adding and subtracting pieces. Each piece is a term: whatever sits between the top-level plus and minus signs. Most terms are a single number, a single letter, or numbers and letters multiplied together, where a letter may carry a power as in x^2. In 5x^2 - 3x + 7 the terms are 5x^2, -3x and 7.
The sign in front belongs to the term. Subtracting a number is the same as adding its opposite, so
and every expression can be read as a sum. The terms are the things being added, so the second term is -3x, not 3x.
Only the plus and minus signs at the top level split an expression into terms. A sign inside brackets does not: in 4(x + 1) + y the terms are 4(x + 1) and y, because 4(x + 1) is one product, 4 times the number x + 1.
The coefficient of a term is its number part: the number that multiplies the letters. The coefficient of 5x^2 is 5, and the coefficient of -3x is -3. The power is not part of the coefficient; the 2 in 5x^2 belongs to the letter part x^2.
A term with no letter in it is a constant term, or just a constant. In 5x^2 - 3x + 7 the constant is 7. It is called constant because its value stays the same whatever numbers the letters stand for.
Terms are like terms when they have exactly the same letters, each raised to the same power. Only the coefficients may differ. So 3x and -8x are like terms, 3x and 3x^2 are not, and 2ab and 5ba are, because ba and ab are the same product. All constants are like terms with one another.
| Term | Letter part | Coefficient |
|---|---|---|
| 5x^2 | x^2 | 5 |
| -3x | x | -3 |
| 7 | none: a constant |
Intuition
Give each letter part its own shape of tile. A large square with side x has area x^2, a long strip x by 1 has area x, and a small square 1 by 1 has area 1. The expression 2x^2 + 3x + 4 is then two large squares, three strips and four small squares.
Now each name has a picture. The letter part says which shape. The coefficient says how many tiles of that shape. The constant is the pile of small squares, the one part you can describe without mentioning x at all.
Like terms are tiles of the same shape, which is why they can be counted together: three strips and five strips are eight strips. Unlike terms are different shapes, and no number of strips makes a large square. In the figure the side x is deliberately not a whole number of small squares, since x could be any length.
Imagine x getting longer. The large squares and the strips grow; the small squares do not change at all. That is what makes the constant constant.
Results
Every expression is a sum of its terms. Rewriting each subtraction as adding the opposite,
so each term carries the sign written in front of it.
Hidden coefficients. A letter written alone has coefficient 1, a letter with only a minus sign has coefficient -1, and a letter divided by a number has a fraction as its coefficient:
The constant is the value at zero. Write an expression with no brackets, with every letter raised only to a positive whole-number power. Then its constant terms add up to the value of the expression when every letter is 0.
Like terms add by adding their coefficients. For any letter part, here x,
Unlike terms cannot be merged this way. Doing this across a whole expression is combining like terms.
Why it is true
Subtraction as adding. Taking away b and adding -b move you the same distance in the same direction along the number line, so a - b and a + (-b) always land on the same number.
Hidden coefficients. Multiplying by 1 changes nothing, so x = 1 \cdot x. The opposite of a number is that number times -1, so -x = -1 \cdot x. Dividing by 4 is the same as multiplying by \displaystyle \frac{1}{4}, so \displaystyle \frac{x}{4} = \frac{1}{4} \cdot x.
The value at zero. Every term with a letter has that letter as a factor. A power such as x^2 = x \cdot x does too, because the power is a positive whole number. Put 0 in for every letter and each of those terms becomes a product with a factor of 0, which is 0. Only the constants are left, so the value is their sum.
Adding like terms. The distributive property says (3 + 5)x = 3x + 5x. Read from right to left, it says 3x + 5x = 8x. The same works with x^2 or ab as the shared letter part. With 3x + 5x^2 there is no shared letter part to pull out, so the two terms stay separate.
Worked examples
Three terms, nothing hidden
Name the terms of 6m + 4n^2 + 11. Give the coefficient of each term that has a letter, and name the constant.
SolutionThe expression has two plus signs at the top level, so it splits into three pieces. The terms are 6m, 4n^2 and 11.
The term 6m is 6 times m, so the coefficient of 6m is 6. The term 4n^2 is 4 times n^2, so the coefficient of 4n^2 is 4. The 2 is the power on n, part of the letter part, not the coefficient.
The term 11 has no letter. The constant is 11.
Minus signs travel with their terms
Name the terms, the coefficients and the constant of 7y^2 - 4y - 10.
SolutionRead each subtraction as adding the opposite:
Now the pieces being added are plain to see. The terms are 7y^2, -4y and -10. The coefficients are 7 and -4, and the constant is -10.
Check. The constant should be the value at y = 0: 7(0)^2 - 4(0) - 10 = -10. It is.
Coefficients you cannot see
Give the coefficient of each term of \dfrac{p}{4} - q + 0.6r, and say what its constant term is.
SolutionThe terms are \displaystyle \frac{p}{4}, -q and 0.6r. None of them shows a number written in front, so each coefficient has to be read off.
Dividing by 4 is multiplying by \displaystyle \frac{1}{4}, so \displaystyle \frac{p}{4} = \frac{1}{4}p. The term -q is -1 times q. The term 0.6r is 0.6 times r. The coefficients are \displaystyle \frac{1}{4}, -1 and 0.6.
Every term has a letter, so there is no constant written. It is fair to say the constant term is 0: putting p = q = r = 0 gives 0 - 0 + 0 = 0.
Brackets make one term
Look at 3(x + 4) - 2xy + 5. How many terms does it have as written? What is the coefficient of the xy term? Rewrite it without brackets, put like terms together, and name the constant term.
SolutionThe plus sign inside the brackets does not count, because it is not at the top level. The top-level signs split the expression into 3(x + 4), -2xy and 5, so it has three terms.
The term -2xy is -2 times the product xy, so its coefficient is -2.
By the distributive property, 3(x + 4) = 3x + 12, so
Written this way there are four terms, and two of them, 12 and 5, are constants. Constants are like terms, and 12 + 5 = 17, so the expression is 3x - 2xy + 17, and its constant term is 17.
Check. At x = 0 and y = 0 the original gives 3(4) - 0 + 5 = 17. Notice that before the brackets were removed, the only constant you could see was 5; the bracket was hiding the 12. That is why the value-at-zero result asks for an expression with no brackets.
Sorting like terms
Group the like terms in 6x^2 + 5x - 2x^2 + 9 - x + 4xy - 3, then use the groups to write the expression with fewer terms.
SolutionFirst list the terms, each with its sign: 6x^2, 5x, -2x^2, 9, -x, 4xy and -3. Then sort them by letter part.
- Letter part x^2: 6x^2 and -2x^2.
- Letter part x: 5x and -x.
- Letter part xy: 4xy alone. It has an x, but it also has a y, so it is not like 5x.
- No letter: 9 and -3.
Like terms add by adding coefficients: 6 + (-2) = 4, 5 + (-1) = 4 and 9 + (-3) = 6. So the expression is
Seven terms have become four. Three of them now have the coefficient 4, but they are still not like terms: a shared coefficient does not make terms alike, a shared letter part does.
What the numbers mean in a formula
Amara walks dogs. For a walk of h hours she charges C = 12h + 5 dollars. Name the coefficient and the constant, and say what each one means. What does a 3-hour walk cost? If she raises her hourly rate to 14 dollars, which number changes?
SolutionThe terms are 12h and 5. The coefficient is 12 and the constant is 5.
The coefficient tells you what each hour adds. When h goes up by 1, the term 12h goes up by 12, because 12(h + 1) = 12h + 12, and the 5 does not move. So 12 is the price per hour.
The constant is the part of the charge that does not depend on the hours: she charges it for every walk, long or short, like a booking fee. It is also the value at h = 0.
For 3 hours, C = 12(3) + 5 = 41, so the walk costs 41 dollars.
A new hourly rate changes the coefficient and nothing else. The formula becomes C = 14h + 5.
Finding a coefficient and a constant from two values
An expression has the form ax + b, where a and b are fixed numbers. Its value is 7 when x = 0 and 11 when x = 2. Find a and b.
SolutionUse the value at zero first. When x = 0 the term ax is 0, so the value is just the constant. That gives b = 7.
Now use x = 2. The expression is 2a + 7, and it equals 11:
so a = 2, and the expression is 2x + 7.
Check. 2(0) + 7 = 7 and 2(2) + 7 = 11.
Here a and b are letters, but they are constants: each stands for one fixed number, while x is free to change. Which job a letter does depends on the sentence around it, as in what a letter stands for.
Boundaries
The terms depend on how the expression is written. 2(x + 1) has one term and 2x + 2 has two, yet they are equal for every x. Counting terms is a fact about the written form, not about the number it stands for.
A letter can be a constant. In ax + b the letters a and b name fixed numbers, so a is a coefficient and b is the constant. The same goes for \pi in \pi r^2: \pi is a number, so the coefficient of \pi r^2 is \pi.
“The coefficient of x” can mean two things. In this lesson the coefficient of 6xy is its number part, 6. Later, when one letter is the one you care about, “the coefficient of x” in 6xy often means everything else that multiplies x, which is 6y. The sentence tells you which is meant.
Some books give a constant a coefficient. They say the coefficient of 7 is 7, thinking of 7 as 7 times no letter at all. Both habits agree about every term that has a letter.
Like terms need the same powers, not just the same letters. x^2y is x \cdot x \cdot y and xy^2 is x \cdot y \cdot y, so they are not like terms, although they use the same two letters. The order the letters are written in does not matter, and neither do the coefficients.
The value-at-zero result has conditions. It needs no brackets and only positive whole-number powers. \displaystyle \frac{6}{x} + 1 has constant 1, but it has no value at all at x = 0. 2^x + 3 has constant 3, but at x = 0 it is 2^0 + 3 = 4, because x there is a power, not a factor.
Where this goes next
Like terms are what make expressions shorter. Putting them together, with all the signs handled, is combining like terms, and nearly every later piece of algebra does it on the way.
A single term, a number times letters with powers, is a monomial, and multiplying monomials works term by term: multiply the coefficients, add the powers. A sum of such terms is a polynomial, and its biggest power and the coefficient that goes with it decide most of its behavior, which is where the degree of a polynomial starts.
The coefficient and the constant of ax + b return as soon as you graph a straight line. The coefficient is the rate of change, the amount added for each step in x, as in Amara’s hourly price, and the constant is where the line starts, as in slope-intercept form.
Finally, terms are parts of expressions, and an expression is not yet a claim. Putting two expressions on either side of an equals sign makes something that can be true or false, which is the difference between an expression and an equation.
Exercises
Naming the terms
- List the terms of 8p + 3q + 1.
- List the terms of 2n^2 - 7n + 6.
- List the terms of -a + 4b - 9.
- How many terms does 5 + 2(k - 1) + k^2 have? Name them.
Coefficients and constants
- Give the coefficient of each term of 9t - u + \dfrac{v}{2}.
- What is the constant term of 4c^2 - 11 + c?
- What is the constant term of 3m - 8n?
- In 12 - y + 0.4y^2, give the coefficient of the y^2 term and of the y term.
- What is the coefficient of the term -\dfrac{3ab}{5}?
- The expression \pi r^2 + 2\pi r has one term in r^2 and one in r. Give the coefficient of each term.
Like terms
- Sort these terms into groups of like terms: 5y, 2, y^2, -8y, 13, 6y^2, yz.
- Group the like terms in 3h^2 + h - 7h^2 + 4 + 2h - 1.
- Are 4x^2y and 4xy^2 like terms? Explain.
- Are -6pq and 9qp like terms? Explain.
- In 10a - 3ab + 2b - ab + a, which terms are like terms, and which term has no like term?
Building and reading expressions
- Write an expression with three terms in which the coefficient of x^2 is 1, the coefficient of x is -4 and the constant is 9.
- A gym charges C = 15m + 40 dollars for m months. Which number is the joining fee and which is the monthly charge? What do 6 months cost?
- Write 5(y + 2) - 3 without brackets, put like terms together, and name the terms you end with.
- The expression kx + 5 has the value 26 when x = 3. Find k.
- An expression of the form ax + b has the value -2 when x = 0 and the value 3 when x = 1. Find a and b.
Reasoning
- Explain why the coefficient of x is 1 and not 0.
- Jonas says the terms of 2x - 5 are 2x and 5. What has he missed, and why does it matter?
- When x = 2, both x^2 and 2x equal 4. Explain why they are still not like terms.
- Explain why putting x = 0 gives the constant term of 3x^2 - x + 8, but not the constant term of 2^x + 3.
- Can two expressions that are equal for every value of x have different numbers of terms? Give an example or explain why not.
Answers
- 8p, 3q and 1. The two top-level plus signs split it into three pieces.
- 2n^2, -7n and 6. The minus sign belongs to the term after it.
- -a, 4b and -9. Reading it as (-a) + 4b + (-9) shows every sign.
- Three: 5, 2(k - 1) and k^2. The minus sign inside the brackets is not at the top level, so 2(k - 1) is a single term.
- 9, -1 and \displaystyle \frac{1}{2}. The term -u is -1 times u, and \displaystyle \frac{v}{2} = \frac{1}{2}v.
- -11. It is the only term with no letter, and the minus sign in front belongs to it. Check: at c = 0 the value is 0 - 11 + 0 = -11.
- Every term has a letter, so the constant term is 0. At m = 0 and n = 0 the value is 0.
- The y^2 term 0.4y^2 has coefficient 0.4, and the y term -y has coefficient -1.
- \displaystyle -\frac{3}{5}, because \displaystyle -\frac{3ab}{5} = -\frac{3}{5}ab.
- \pi and 2\pi. The number \pi multiplies r^2 in the first term, and 2\pi multiplies r in the second; \pi is a number, not a variable.
- 5y and -8y; y^2 and 6y^2; 2 and 13; yz on its own. Each group shares a letter part, and yz is the only term with both y and z.
- 3h^2 and -7h^2; h and 2h; 4 and -1. The last two are like because all constants are.
- No. 4x^2y = 4 \cdot x \cdot x \cdot y and 4xy^2 = 4 \cdot x \cdot y \cdot y. They use the same letters, but x and y carry different powers.
- Yes. qp and pq are the same product, since the order of multiplying does not matter, so the letter parts match. Only the coefficients, -6 and 9, differ.
- 10a and a are like terms, and -3ab and -ab are like terms. 2b has no like term.
- x^2 - 4x + 9, with its terms in any order. The term x^2 has the hidden coefficient 1.
- The constant 40 is the joining fee and the coefficient 15 is the monthly charge. Six months cost 15(6) + 40 = 130, which is 130 dollars.
- 5(y + 2) - 3 = 5y + 10 - 3 = 5y + 7, so the result is 5y + 7 and its terms are 5y and 7.
- k = 7. At x = 3 the value is 3k + 5 = 26, so 3k = 21.
- b = -2 and a = 5, so the expression is 5x - 2. At x = 0 only the constant is left, so b = -2; at x = 1 the value is a + (-2) = 3.
- A coefficient is the number that multiplies the letters. x is 1 \cdot x, so its coefficient is 1. A coefficient of 0 would mean 0 \cdot x = 0, and the term would not be there at all.
- The minus sign. The expression is 2x + (-5), so the terms are 2x and -5. It matters because the constant is -5, not 5: at x = 0 the value is -5, and taking the term as 5 would give the wrong value and the wrong sum when like terms are put together.
- Like terms are decided by the letter part, not by the value at one number. The letter parts here are x^2 and x, and those are different: they are equal only at x = 0 and x = 2, not for every x. At x = 3, x^2 = 9 but 2x = 6. In tiles, one is a square of side x and the other two strips x by 1, and no count of strips makes a square.
- In 3x^2 - x + 8 each term with a letter has x as a factor, so at x = 0 those terms are 0 and the value is the constant 8. In 2^x + 3 the x is a power, not a factor, and 2^0 = 1, so the value at x = 0 is 4, while the constant term is 3.
- Yes. 2(x + 1) has one term and 2x + 2 has two, and by the distributive property they are equal for every x. The number of terms belongs to the way an expression is written.