06-Polynomials-4UP

# 06-Polynomials-4UP - Polynomials Polynomials& MATLAB...

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Unformatted text preview: Polynomials Polynomials & MATLAB Taylor polynomials Polynomials Dhavide Aruliah UOIT MATH 2070U c D. Aruliah (UOIT) Polynomials MATH 2070U 1 / 22 Polynomials Polynomials & MATLAB Taylor polynomials Polynomials 1 Polynomials 2 Polynomials in MATLAB 3 Taylor’s Theorem and Taylor polynomials c D. Aruliah (UOIT) Polynomials MATH 2070U 2 / 22 Polynomials Polynomials & MATLAB Taylor polynomials Polynomials Monomials: x ≡ 1, x 1 ≡ x , x 2 , x 3 , x 4 , . . . Polynomials: linear combinations of monomials f ( x ) = n ∑ k = a n- k x n- k = a n x n + a n- 1 x n- 1 + ··· + a 1 x + a (assume coefficients a , a 1 , . . . , a n ∈ R or C ) deg ( f ( x )) = degree of f ( x ) = largest exponent in f ( x ) Leading coefficient: coefficient a n of highest degree monomial c D. Aruliah (UOIT) Polynomials MATH 2070U 4 / 22 Polynomials Polynomials & MATLAB Taylor polynomials Classifications of polynomials Degree Name Example constant f ( x ) = 4 1 linear f ( x ) = 3 x- 2 2 quadratic f ( x ) = x 2- x + 4 3 cubic f ( x ) = 4 x 3- x 2 4 quartic f ( x ) = 6 x 4 + ( 3/2 ) x 3- 7 x 2- 5 x- 3/7 5 quintic f ( x ) =- 2 x 5 + 4 x 4- x 3 + x 2 /2- 4 x + 7 For degree n > 3, preferred terminology “polynomial of degree n ” Binomial: exactly 2 nonzero terms (e.g., x 3- 4 x ) Trinomial: exactly 3 nonzero terms (e.g., 3 x 4- x 2 + 1) Monic polynomial: leading coefficient a n = 1 c D. Aruliah (UOIT) Polynomials MATH 2070U 5 / 22 Polynomials Polynomials & MATLAB Taylor polynomials Examples f ( x ) = 5 x 2- 3 x + 2 g ( x ) = 7 x 4- x 3 + 2 x 2 + 4 x- 6.1 h ( x ) =- x 3 + x 2 p ( x ) = x + 2 deg ( f ( x )) = 2 and the leading coefficient is 5 (quadratic) deg ( g ( x )) = 4 and the leading coefficient is 7 (quartic) deg ( h ( x )) = 3 and the leading coefficient is -1 (cubic) deg ( p ( x )) = 1 and the leading coefficient is 1 (linear, monic) c D. Aruliah (UOIT) Polynomials MATH 2070U 6 / 22 Polynomials Polynomials & MATLAB Taylor polynomials Adding, subtracting, multiplying polynomials Sum of f ( x ) + g ( x ) polynomials f ( x ) & g ( x ) is a polynomial e.g., f ( x ) = x 3 + 1, g ( x ) = 2 x- 3 ⇒ f ( x ) + g ( x ) = x 3 + 2 x- 2 Difference f ( x )- g ( x ) computed by f ( x ) + [- g ( x )] e.g., f ( x ) = x 3 + 1, g ( x ) = x 3- 3 ⇒ f ( x )- g ( x ) = 4 Notice deg ( f ( x ) + g ( x )) ≤ max ( deg...
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06-Polynomials-4UP - Polynomials Polynomials& MATLAB...

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