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4.1 Polynomial functions Day2

4.1 Polynomial functions Day2. Objectives. Use long division and synthetic division to find zeroes Apply the remainder and factor theorem. Let f(x) and d(x) be polynomials with degree of f greater than or equal to degree of d, and d(x) not equal to 0. Then.

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4.1 Polynomial functions Day2

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  1. 4.1 Polynomial functions Day2 • Objectives. • Use long division and synthetic division to find zeroes • Apply the remainder and factor theorem

  2. Let f(x) and d(x) be polynomials with degree of f greater than or equal to degree of d, and d(x) not equal to 0. Then.... Use polynomial long division to divide: Answer: Remainder

  3. Factor Theorem When the remainder is zero, then the divisor is a factor of the polynomial. Division may be used to help you start factoring a polynomial! y = x3 + 3x2 – 4 Is x-1 a factor of Yes it is a factor. (x-1) Factor the remaining polynomial. (x-1)(x+2)(x+2)

  4. Synthetic Division: Shortcut used when dividing by a linear divisor. (2x3+x2-4x+3)÷(x+1) Divide by the zero. x+1=0, then x=-1 And only use coefficients! 2 1 -4 3 ______________ -1 -2 1 3 Multiply by the zero and add 2 -1 -3 6 remainder answer: Divisor

  5. Fundamental Connections if the remainder is zero when dividing by (x-k) where k is a real number: 1. x = kis a solution or root of the equation f(x)=0 2. k is a zero of the function. 3. k is an intercept of the graph 4. x-k is a factor of f(x) (factor theorem)

  6. Use the factor theorem to show that x+1 is a factor of f(x)=x25+1 but not of g(x)=x25-1 If x+1 is a factor then x=-1 is a zero. f(-1)= -1+1=0 yes, since x =-1 is a zero, x+1 is a factor g(-1)=-1-1=-2 No, -1 is not a zero, therefore not a factor

  7. Remainder theorem If the remainder is not equal to 0, then the remainder theorem states that f(k) =r Find f(-1) for f(x)=3x 2+7x+20 using the remainder theorem. Divide by x+1 using synthetic division. The remainder is the answer. -1 3 7 20 ______________ 3 4 16 -3 -4 remainder Therefore f(-1)=16

  8. Given: 3, -3, and 4 are zeroes....find an equation. If these are the zeroes...then what are the factors... y = (x-3)(x+3)(x-4) then foil...foil....

  9. Lesson Close Name three ways to find the zeroes of a polynomial function

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