# Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms

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1 Polynomials

2 Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term")... so it says "many terms

3 Polynomials A polynomial looks like this:

4 Term A number, a variable, or the product/quotient of numbers/variables.

5 A term has 3 components: 5x 3 Exponent: Can only be 0,1,2, 3, Coefficient: can by any real number including zero. Variable: Usually denoted as x

6 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

7 Remember

8 These are more examples of polynomials 3x x - 2-6y 2 - ( 7 / 9 )x 3xyz + 3xy 2 z - 0.1xz - 200y v w 5 5 (which is really 5x 0 )

9 A polynomial can have: Terms that can be combined by adding, subtracting or multiplying. A Polynomial can NOT have a division by a variable in it.

10 These are not allowed as a term 2 x+2 1 x in a polynomial is not, because dividing by a variable is not allowed is not either 3xy -2 is not, because : y 2 = 1 y 2 x is not, because the exponent is "½"

11 These are allowed as a term in a polynomial 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 (= etc)

12 Monomial, binomial, and trinomial

13 There is also quadrinomial (4 terms) and quintinomial (5 terms), but those names are not often used. Can Have Lots and Lots of Terms Polynomials can have as many terms as needed, but not an infinite number of terms.

14 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 x 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

15 General Term of a polynomial

16 Degree of a Term The exponent of the variable. We will find them only for one-variable terms.

17 Term 3 4x -5x 2 18x 5 Degree of Term

18 Polynomial A term or the sum/difference of terms which contain only 1 variable. The variable cannot be in the denominator of a term.

19 Degree of a Polynomial The degree of the term with the highest degree.

20 Polynomial Degree of Polynomial 6x 2-3x x + 5x x x 1 x 3 + x 2 + x + 1 3

21 Standard Form of a Polynomial A polynomial written so that the degree of the terms decreases from left to right and no terms have the same degree.

22 Not Standard 6x + 3x x + 5x 4 x x 1 + x 2 + x + x 3 Standard 3x 2 + 6x - 2 5x 4-4x x + 10 x 3 + x 2 + x + 1

23 Naming Polynomials Polynomials are named or classified by their degree and the number of terms they have.

24 Polynomial 7 5x + 2 4x 2 + 3x - 4 6x 3-18 Degree Degree Name constant linear quadratic cubic For degrees higher than 3 say: of degree n or n th degree x 5 + 3x x of degree 5 or 5 th degree of degree 8 or 8 th degree

25 Polynomial 7 5x + 2 4x 2 + 3x - 4 6x 3-18 # Terms # Terms Name monomial binomial trinomial binomial For more than 3 terms say: a polynomial with n terms or an n-term polynomial 11x 8 + x 5 + x 4-3x 3 + 5x 2-3 a polynomial with 6 terms or a 6-term polynomial

26 Polynomial Name -14x 3-1.2x 2-1 7x - 2 3x 3 + 2x - 8 2x 2-4x + 8 x cubic monomial quadratic monomial constant monomial linear binomial cubic trinomial quadratic trinomial 4 th degree binomial

27 Adding and Subtracting Polynomials To add or subtract polynomials, simply combine like terms.

28 (5x 2-3x + 7) + (2x 2 + 5x - 7) = 7x 2 + 2x (3x 3 + 6x - 8) + (4x 2 + 2x - 5) = 3x 3 + 4x 2 + 8x - 13

29 (2x 3 + 4x 2-6) (3x 3 + 2x - 2) (2x 3 + 4x 2-6) + (-3x x - -2) = -x 3 + 4x 2-2x - 4 (7x 3-3x + 1) (x 3-4x 2-2) (7x 3-3x + 1) + (-x x ) = 6x 3 + 4x 2-3x + 3

30 TRY IT THE VERTICAL WAY! 7y 2 3y y 2 + 3y 4 15y 2 + 0y + 0 = 15y 2 2x 3 5x 2 + 3x 1 (8x 3 8x 2 + 4x + 3) 6x 3 + 3x 2 x 4

31 (7y 3 +2y 2 + 5y 1) + (5y 3 + 7y) 7y 3 + 2y 2 + 5y 1 + 5y 3 + 0y 2 + 7y y 3 + 2y y 1

32 (b b + 1) + (8b 4 + 2b 3b 2 ) Rewrite in standard form! b 4 + 0b 3 + 0b 2 + 5b 5 + 8b 4 + 0b 3 3b 2 + 2b + 0 9b 4 + 0b 3 3b 2 + 7b 5 = 9b 4 3b 2 + 7b 5

33 Polynomials: Algebra Tiles Algebra tiles are tools that help one represent a polynomial.

34 Polynomials: Algebra Tiles What polynomial is represented below?

35 Polynomials: Multiplying by a Monomial w/tiles When you multiply, you are really finding the areas of rectangles. The length and width are the factors and the product is the area. For example multiply 2x (3x + 1) 3x+1 6x 2 + 2x 2x

36 Multiplying Polynomials Distribute and FOIL

37 Polynomials * Polynomials Multiplying a Polynomial by another Polynomial requires more than one distributing step. Multiply: (2a + 7b)(3a + 5b) Distribute 2a(3a + 5b) and distribute 7b(3a + 5b): 6a ab 21ab + 35b 2 Then add those products, adding like terms: 6a ab + 21ab + 35b 2 = 6a ab + 35b 2

38 Polynomials * Polynomials An alternative is to stack the polynomials and do long multiplication. (2a + 7b)(3a + 5b) (2a + 7b) x (3a + 5b) Multiply by 5b, then by 3a: When multiplying by 3a, line up the first term under 3a. + (2a + 7b) x (3a + 5b) 21ab + 35b 2 6a ab Add like terms: 6a ab + 35b 2

39 Polynomials * Polynomials Multiply the following polynomials: 1) x 5 2x 1 2) 3w 2 2w 5 3) 2a 2 a 1 2a 2 1

40 Polynomials * Polynomials 1) x 5 2x 1 (x + 5) x (2x + -1) -x x x 2x 2 + 9x ) 3w 2 2w 5 (3w + -2) x (2w + -5) -15w w w 6w w + 10

41 Polynomials * Polynomials 3) 2a 2 a 1 2a 2 1 (2a 2 + a + -1) x (2a 2 + 1) 2a 2 + a a 4 + 2a a 2 4a 4 + 2a 3 + a + -1

42 Types of Polynomials We have names to classify polynomials based on how many terms they have: Monomial: a polynomial with one term Binomial: a polynomial with two terms Trinomial: a polynomial with three terms

43 F.O.I.L. There is an acronym to help us remember how to multiply two binomials without stacking them. (2x + -3)(4x + 5) F : Multiply the First term in each binomial. 2x 4x = 8x 2 O : Multiply the Outer terms in the binomials. 2x 5 = 10x I : Multiply the Inner terms in the binomials. -3 4x = -12x L : Multiply the Last term in each binomial = -15 (2x + -3)(4x + 5) = 8x x + -12x = 8x x + -15

44 F.O.I.L. Use the FOIL method to multiply these binomials: 1) (3a + 4)(2a + 1) 2) (x + 4)(x - 5) 3) (x + 5)(x - 5) 4) (c - 3)(2c - 5) 5) (2w + 3)(2w - 3)

45 F.O.I.L. Use the FOIL method to multiply these binomials: 1) (3a + 4)(2a + 1) = 6a 2 + 3a + 8a + 4 = 6a a + 4 2) (x + 4)(x - 5) = x x + 4x = x x ) (x + 5)(x - 5) = x x + 5x = x ) (c - 3)(2c - 5) = 2c c + -6c + 15 = 2c c ) (2w + 3)(2w - 3) = 4w w + 6w + -9 = 4w

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