Gifted Precalculus

1
Gifted Precalculus

• Section 2.3 Real Zeros of Polynomial Functions

2
Divide Using Long Division
3
Divide Using Synthetic Division
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The Division Algorithm

• Dividing f(x) by d(x) where the degree of d(x) is
  less than or equal to the degree of f(x) yields
• f(x) d(x)q(x) r(x)
• Dividend (Divisor)(Quotient) Remainder
• If the remainder r(x) is zero, d(x) divides
  evenly into f(x)

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The Remainder Theorem

• If a polynomial f(x) is divided by x - k,the
  remainder is r f(k)

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The Factor Theorem

• A polynomial function f(x) has a factor (x
  k) if and only if f(k) 0

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Express the function in the form f(x) d(x)q(x)
r(x) for the given value of k. Show f(k) r

• f(x) 15x4 10x3 6x2 14 k -2/3

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Use synthetic division to find the function
value. Use your calculator to check.

• f(x) x6 4x4 3x2 2
• Find f(2)

9
Use synthetic division to show that x is a
solution of the polynomial equation. Use the
result to factor the polynomial completely and
list all the real zeros

• 2x3 11x2 18x 9 0 x -3

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The Rational Zero Test

• If the polynomial
• f(x) anxn an-1xn-1 . . . a2x2 a1x a0
  has integer coefficients, every rational zero of
  f has the form
• Rational zero p/q
• where p and q have no common factors other than
  1, p is a factor of the constant term a0 and q is
  a factor of the leading coefficient an

11
Find all the real solutions of the polynomial
function.

• f(x) x3 3x2 x 3
• g(x) 4x4 17x2 4