Polynomials A polynomial is made up of terms that are only added, subtracted or multiplied. A polynomial looks like this:
example of a polynomial this one has 3 terms Polynomial comes form poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms" A polynomial can have: constants (like 3, -20, or ½) variables (like x and y) exponents (like the 2 in y2) but only 0, 1, 2, 3, ... etc That can be combined using: + addition, subtraction, and × multiplication ... but not division! Polynomial or Not?
These are polynomials: • 3x • x-2 • -6y2 - (7/9)x • 3xyz + 3xy2z - 0.1xz - 200y + 0.5 • 512v5+ 99w5 • 1 (Yes, even "1" is a polynomial; it has one term which just happens to be a constant). And these are not polynomials • 2/(x+2) is not, because dividing is not allowed • 1/x is not • 3xy-2 is not, because the exponent is "-2" (exponents can only be 0,1,2,...) • √x is not, because the exponent is "½" But these are allowed: • x/2 is allowed, because it is also (½)x (the constant is ½, or 0.5) • also 3x/8 for the same reason (the constant is 3/8, or 0.375) • √2 is allowed, because it is a constant (= 1.4142...etc)
Monomial, Binomial, Trinomial There are special names for polynomials with 1, 2 or 3 terms:
Can Have Lots and Lots of Terms Polynomials can have as many terms as needed, but not an infinite number of terms. Degree
The degree of a polynomial with only one variable is the largest exponent of that variable. Example: The Degree is 3 (the largest exponent of x) More Examples: The Degree is 1 (a variable without an exponent actually has an exponent of 1) The Degree is 3 (largest exponent of x) The Degree is 5 (largest exponent of x) The Degree is 2 (largest exponent of z) Names of Degrees When you know the degree you can also give it a name! Name Degree 0 Constant 1 Linear 2 Quadratic 3 Cubic 4 Quartic 5 Quintic
Example 7 x+3 x2-x+2 x3-x2+5 6x4-x3+x-2 x5-3x3+x2+8
Example: y = 2x + 7 has a degree of 1, so it is a linear equation Example: 5w2 - 3 has a degree of 2, so it is quadratic Degree of a Polynomial with More Than One Variable If there is more than one variable in the polynomial, you need to look at each term. Terms are separated by + or - signs:
example of a polynomial with more than one variable 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:
• 5xy2 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) • 5y3 has a degree of 3 (y has an exponent of 3) • 3 has a degree of 0 (no variable) The largest degree is 3, so the polynomial has a degree of 3 Writing it Down Instead of saying "the degree of (whatever) is 3" you write it like this:
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: 3x2 - 7 + 4x3 + x6 The highest degree is 6, so that goes first, then 3, 2 and then the constant last: x6 + 4x3 + 3x2 - 7 You don't have to use Standard Form, but it helps.
Adding and Subtracting Polynomials To add polynomials you 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 x2) are the same. In other words, terms that are "like" each other. Note: the coefficients (the numbers you multiply by, such as "5" in 5x) can be different. Example: 7x -2x x are all like terms because the variables are all x Example: (1/3)xy2 -2xy2 6xy2 are all like terms because the variables are all xy2
Adding Polynomials Two Steps: • Place like terms together • Add the like terms Example: Add 2x2 + 6x + 5 and
3x2 - 2x - 1
Start with: 2x2 + 6x + 5 Place like terms together: 2x2 + 3x2 + 2 Add the like terms: (2+3)x + = 5x2 + 4x + 4
+ 3x2 - 2x - 1 6x - 2x + (6-2)x +
5-1 (5-1)
Adding Several Polynomials You can add several polynomials together like that. Example: Add (2x2 + 6y + 3xy), (3x2 - 5xy - x) and (6xy + 5) Line them up in columns and add: 2x2 + 6y + 3xy 3x2 - 5xy - x 6xy + 5 5x2 + 6y + 4xy - x + 5 Using columns helps you to match the correct terms together in a complicated sum.
Subtracting Polynomials To subtract Polynomials, first reverse the sign of each term you are subtracting (in other words turn "+" into "-", and "-" into "+"), then add as usual.
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:
(3 xy ).(5 x 2 y 3 z ) = 15 x3 y 4 z (Note: I used "·" to mean multiply. In Algebra we don't like to use "×" because it looks too much like the letter "x") 1 term × 2 terms (monomial times binomial) Multiply the single term by each of the two terms, like this:
2 x.( x + 3 xy ) = 2 x 2 + 6 x 2 y (I did that one a bit faster by multiplying in my head before writing it down) 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 differrent multiplications ... Why? Matching up Partners Two friends (Alice and Betty) challenge two other friends (Charles and David) to individual tennis matches.
How many matches does that make? • Alice needs to play Charles, and then Alice needs to play David. • And Betty has to play Charles and then Betty has to play David. They could play in any order, so long as each of the first two friends gets to play each of the second two friends. It is the same when we are multiplying binomials! instead of Alice and Betty, let's just use a and b, and Charles and David can be c and d:
(a + b).(c + d ) = ac + ad + bc + bd You can multiply them in any order so long as each of the first two terms gets multiplied by each of the second two terms. But there is a handy way to help you remember to multiply each term called "FOIL" that stands for "Firsts, Outers, Inners, Lasts":
• • • •
Firsts: ac Outers: ad Inners: bc Lasts: bd
So you multiply the "Firsts" (the first terms of both polynomials), then the "Outers", etc. 2 terms × 3 terms (binomial times trinomial) "FOIL" won't work here, because there are more terms now. But just remember: Multiply each term in the first polynomial by each term in the second polynomial And always remember to add Like Terms. Example: (x + 2y)(3x – 4y + 5)= 3x2 – 4xy + 5x + 6xy – 8y2 + 10y= 3x2 + 2xy + 5x – 8y2 + 10y I added -4xy and 6xy because they are Like Terms. Note: 6yx means the same thing as 6xy Exercises: Classify as a monomial, binomial, or trinomial: 2.- -14x2 - 14x + 19 1.- 9x – 4 Monomial Binomial Trinomial Add 3.- (14x + 15) + (10x + 14) 4.- (8x + 10) + (11x + 10)
Monomial
Binomial
Trinomial
5.- (14x2 + 5x + 19) + (11x2 + 18x + 17)
Subtract 6.- (7x + 7) - (11x + 13
7.- (11x + 17) - (14x + 18)
8.- (7x2 + 17x + 13) - (12x2 + 10x + 4)
Multiply 9.- (6x) (11x + 9)
10.- 5x (4x + 1)
11.- 10x (11x2 + 5x + 4)
Use FOIL 12.- (11x + 8) (2x + 5)
13.- (7x + 7) (5x + 3)
14.- (5x + 2) (11x + 7)
Multiply 15.- (5x2 + 6x - 10) (8x - 3)
16.- (9x2 + 11x - 9) (12x - 5)
17.- (-7x7 - 12x6 + 9) (6x2 - 10x - 7)
Add, Subtract, and Multiply 18.- [(7x + 4) (-8x - 8)] + [(8x + 9) (5x + 6)]
19.- [(11x) (10x + 11)] - [(16x + 14) - (10x + 16)]