Polynomials and Polynomial Equations A Polynomial is any expression that has constants, variables and exponents, and can be combined using addition, subtraction, multiplication and division, but: no division by a variable. a variable's exponents can only be 0,1,2,3,... etc. it can't have an infinite number of terms. Polynomial comes from the Greek words poly- (meaning "many") and -nomial (meaning "name")... so it says "many names" in our case many terms A polynomial can have: constants (like 3, -20, or ½) variables (like x and y) exponents (like the 2 in y 2 ), but only 0, 1, 2, 3,... etc are allowed that can be combined using addition, subtraction, multiplication and division...... except...... not division by a variable (so something like 2/x is out) So: A polynomial can have constants, variables and exponents, but never division by a variable These are polynomials: 3x x - 2-6y2 - (7/9)x 3xyz + 3xy2z - 0.1xz - 200y + 0.5 512v5+ 99w5 5 (Yes, even "5" is a polynomial, one term is allowed, and it can even be just a constant!)
And these are not polynomials 3xy-2 is not, because the exponent is "-2" (exponents can only be 0,1,2,...) 2/(x+2) is not, because dividing by a variable is not allowed 1/x is not either x is not, because the exponent is "½" (see fractional exponents) But these are allowed: x/2 is allowed, because you can divide by a constant also 3x/8 for the same reason 2 is allowed, because it is a constant (= 1.4142...etc) Monomials, Binomials and Trinomials Definition of a monomial: A monomial is a variable, a real number, or a multiplication of one or more variables and a real number with whole-number exponents Examples of monomials and non-monomials Monomials 9 x 9x 6xy 0.60x 4 y Not monomials y - 6 x -1 or 1/x (x) or x 1/2 6 + x a/x Polynomial definition: A polynomial is a monomial or the sum or difference of monomials. Each monomial is called a term of the polynomial. Polynomials can have many terms, but not and infinite number of terms. Important!:Terms are separated by addition signs and subtraction signs, but never by multiplication signs A polynomial with one term is called a monomial A polynomial with two terms is called a binomial A polynomial with three terms is called a trinomial Examples of polynomials: Polynomial Number of terms Some examples Monomial 1 2, x, 5x 3 Binomial 2 2x + 5, x 2 - x, x - 5
Trinomial 3 x 2 + 5x + 6, x 5-3x + 8 Variables: A polynomial may have more than one variable. For example, x + y and x 2 + 5y + 6 are still polynomials although they have two different variables x and y. By the same token, a monomial can have more than one variable. For example, 2xyz is a monomial. So, a polynomial can have no variables at all, like 21 is a polynomial. It has just one term, which is a constant. Or one variable: x 4-2x 2 +x has three terms, but only one variable (x). Or two or more variables: xy 4-5x 2 z has two terms, and three variables (x, y and z) Exercices For the expressions below, look for all expressions that are polynomials. For those that are polynomials, state whether the polynomial is a monomial, a binomial, or a trinomial 1) 3.4 + 3.4x 2) z 2 + 5z -1 + 6 3) -8 4) 2c 2 + 5b + 6 5) 14 + x 6) 5x - 2-1 7) 4 b 2-2 b -2 8) f 2 + 5f + 6 Answer: 1), 3), 4), 5), 6), and 8) are polynomials. 1), 5), and 6) are binomials. 3) is a monomial. 4) and 8) are trinomials. 2) and 7) are not because they have negative exponents Notice that 6) is still a polynomial although it has a negative exponent. It is because it is the exponent of a real number, not a variable. In fact, 5x - 2-1 = 5x + 1/2 = 5x + 0.5 What is Special about Polynomials? Because of the strict definition, polynomials are easy to work with. For example we know that: If you add polynomials you get a polynomial If you multiply polynomials you get a polynomial So you can do lots of additions and multiplications, and still have a polynomial as the result.
Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines. Example: x 4-2x 2 +x See how nice and smooth the curve is? Degree (You can also divide polynomials but the result may not be a polynomial) In Algebra "Degree" is sometimes called "Order". The degree of a polynomial with only one variable is the largest exponent of that variable. Example: 4x 3 - x + 3 The Degree is 3 (the largest exponent of x) More Examples: 4x The Degree is 1 (a variable without an exponent actually has an exponent of 1) 4x 3 - x + 3 The Degree is 3 (largest exponent of x) 2x 5 2x 2 + x The Degree is 5 (largest exponent of x) z 2 z + 3 The Degree is 2 (largest exponent of z) Names of Degrees When we know the degree we can also give it a name! Degree Name Example 0 Constant 7 1 Linear x+3 2 Quadratic x 2 x+2 3 Cubic x 3 x 2 +5 4 Quartic 6x 4 x 3 +x 2 5 Quintic x 5 3x 3 +x 2 +8 example: y = 2x + 7 has a degree of 1, so it is a linear equation example: 5w 2 3 has a degree of 2, so it is quadratic equation Degree of a Polynomial with More Than One Variable When a polynomial has more than one variable, we need to look at each term. Terms are separated by + or - signs:
For each term: Find the degree by adding the exponents of each variable in it. The largest such degree is the degree of the polynomial. Example: what is the degree of this polynomial: 5xy 2 3x + 5y 3 3 Checking each term: 5xy 2 has a degree of 3 (x has an exponent of 1, y has 2, and 1+2=3) 3x has a degree of 1 (x has an exponent of 1) 5y 3 has a degree of 3 (y has an exponent of 3) 3 has a degree of 0 (no variable) The largest degree of those is 3, so the polynomial has a degree of 3 Standard Form The Standard Form for writing a polynomial is to put the terms with the highest degree first. Example: Put this in Standard Form: 3x 2-7 + 4x 3 + x 6 The highest degree is 6, so that goes first, then 3, 2 and then the constant last: x 6 + 4x 3 + 3x 2-7 (You don't have to use Standard Form, but it is convention and it helps when working problems.) Adding and Subtracting Polynomials A polynomial looks like this: this one has 3 terms To add polynomials we simply add any like terms together.. so what is a like term? Like Terms Like Terms are terms whose variables (and their exponents such as the 2 in x 2 ) are the same. In other words, terms that are "like" each other. 7x x -2x πx are all like terms because the variables are all x Note: the coefficients (the numbers you multiply by, such as "7" in 7x) can be different. Are these like terms? (1/3)xy 2-2xy 2 6xy 2 xy 2 /2 YES! They all have the variables xy 2
Adding Polynomials Two Steps: Place like terms together Add the like terms Example: Add 2x 2 + 6x + 5 and 3x 2-2x - 1 Start with: 2x 2 + 6x + 5 + 3x 2-2x - 1 Place like terms together: 2x 2 + 3x 2 + 6x - 2x + 5-1 Add the like terms: (2+3)x 2 + (6-2)x + (5-1) = 5x 2 + 4x + 4 What mathematical Law did you use that also proves we can do this? You can add polynomials together by inspection, or you can line them up by like term and add like this: Example: Add (2x 2 + 6y + 3xy), (3x 2-5xy - x) and (6xy + 5) Line them up in columns and add: 2x 2 + 6y + 3xy 3x 2-5xy - x 6xy + 5 5x 2 + 6y + 4xy - x + 5 Subtracting Polynomials To subtract Polynomials, first reverse the sign of each term we are subtracting (in other words turn "+" into "-", and "-" into "+"), then add as usual. Example: subtract (5y 2 + 2xy - 9) - (2y 2 + 2xy 3) use the Distributive property to change the second polynomial like this: -2y 2-2xy + 3 then add the two: 5y 2 + 2xy - 9 + -2y 2-2xy + 3 = 3y 2-6
Multiplying Polynomials To multiply two polynomials multiply each term in one polynomial by each term in the other polynomial, add those answers together, and simplify if needed. Let us look at the simplest cases first. 1 term 1 term (monomial times monomial) To multiply one term by another term, first multiply the constants, then multiply each variable together and combine the result, like this: (2xy)(4y) = (2 4)(xy y) (Commutative property) = (8)(xy 2 ) = 8xy 2 (standard form) 1 term 2 terms (monomial times binomial) Multiply the single term by each of the two terms, like this: 2x(x + 3xy) = (2x x) + (2x 3xy) (Distributive property) = 2x 2 = 6x 2 y 2 terms 2 terms (binomial times binomial) Each of the two terms in the first binomial...... is multiplied by...... each of the two terms in the second binomial That is 4 different multiplications! There is a handy way to help us remember to multiply each term called "FOIL". It stands for "Firsts, Outside, Inside, Lasts": Firsts: ac Outsider: ad becomes: Inside: bc ac + ad + bc + bd Last: bd Let us try this on a more complicated example: (2x + 3 )(xz a) Using the Distributive property and FOIL: (2x + 3 )(xz a) = 2x 2 z - 2xa + 3xz - 3a F O I L Use of the Distributive Property allows us to multiply even larger polynomials such as: (x + a + z)(2x + 3y 5) = 2x 2 + 3xy 5x + 2ax + 3ay 5a + 2xz + 3zy 5z There are no like terms so our answer is as simplified as we can make it!
Dividing Polynomials Sometimes it is easy to divide a polynomial by splitting it at the "+" and " " signs, like this: 6x 3 = 6x - 3 = 2x -1 3 3 3 Take a look at this slightly more complicated example: 9x 2 + 12x + 1 = 9x 2 + 12x + 1 = 3x + 4 + 1 3x 3x 3x 3x 3x What happened? The 1st Term had x 2 above and x below, which together becomes just x The 2nd Term had x above and below, so they canceled each other out We couldn't simplify "1/3x" any further. Note: the result is a valid answer but is not a polynomial, because the last term (1/3x) has division by a variable (x). Now, sometimes it helps to remember Distributive, Associative and Commutative Laws and rearrange the top polynomial before dividing, as in this example: 2x 2 + 2x the Distributive law allows us to rearrange to top to 2x(x + 1) x + 1 now divide by (x + 1) and by canceling our answer becomes 2x You can also divide polynomials through long division. That will be a lesson for another time. Now try some of the problems on the next page for more practice with polynomials.