4-1 Polynomial Functions

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Presentation transcript:

4-1 Polynomial Functions

Notes Monomial – one term: 3x Binomial – two terms: 2x + 7 Trinomial – three terms: 3x2 – 4x + 7 Polynomial – many terms: 5x3 – 2x2 + 4x + 8 Degree – largest sum of exponents on a monomial

Examples Tell if a monomial, binomial or trinomial. 7y4 – 4y2 + 2 10x3y2

Examples Find the degree of the polynomial. 3) 3x2 + 5 4) y7 + y5 - 3x4y4 5) p5 + p3m3 + 4m

Zeros of a function: Where f(x) = 0 Imaginary number (i) = √-1 i0- i= i2= i3= I4=

Fundamental theorem of Algebra: for each polynomial with degree n, there are n solutions. (some may be real, some imaginary) Example: x2 has 2 solutions, x3 has three solutions

General Shapes of Polynomials

1) Write a polynomial with roots 4 and 2 2) Write a polynomial with roots 2, 4i, -4i.

State the number of roots in the equation 9x4 -35x2 -4=0. Find them.

Multiplying Polynomials “FOIL” 3) (3x + 3) (2x – 4) 4) (2x -1 ) (x2 + 3x + 1)

Binomial Expansion Remember Pascal's triangle?

(x + y)6 (3x + 2y)4

Dividing Long Division x2 + 3x -8 ÷ x -2

Synthetic Division Make sure there is no leading coefficient. It should be in the form “ x-r” 2) Arrange the coefficients in a line. (If you are missing a term, use 0) 3) Place the number r in front of the box. Use opposite sign 4) Bring down first number 5) Multiply/Add/Multiply/Add

x3 +4x2 -3x-5 ÷ x+3

(x3 –x2+2) ÷ (x + 1)