The Real Zeros of a Polynomial Function

1
The Real Zeros of a Polynomial Function

• 3.6

2
Division Algorithm for Polynomials

• F(x) q(x)g(x)r(x)
• F is the dividend
• Q is the quotient
• G is the divisor
• R is the remainder

3
Remainder Theorem

• Let f be a polynomial function. If f(x) is
  divided by x-c, then the remainder is f(c).

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Find the remainder if

• Pg. 220 example 1

5
Factor Theorem

• Let f be a polynomial function. Then x-c is a
  factor of f(x) if and only f(c)0.
• 1. If f(c)0, then x-c is a factor of f(x).
• 2. If x-c is a factor of f(x), then f(c)0.

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Is _____ a factor? Only

• 12, and 14 on page 231

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Synthetic Division

• Divide 2x3-x22x-3 by (x-1)
• 2 -1 2 -3
• 2 1 3
• 2 1 3 0

x 1
x 1
x 1
8
Synthetic Division

• Divide 2x3-x22x-3 by (x-1)
• (x-1)(2x2x3)
• 2 -1 2 -3
• 2 1 3
• 2 1 3 0

x 1
x 1
x 1
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Synthetic Division

• Divide -3x3x2-5 by (x1)
• -1 -3 1 0 -5
• 3 -4 4
• -3 4 -4 -1

x -1
x -1
x -1
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Synthetic Division

• Divide -3x3x2-5 by (x1)
• (x1)(-3x24x-4) -1
• -1 -3 1 0 -5
• 3 -4 4
• -3 4 -4 -1

x -1
x -1
x -1
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Write in Factored form

• 12, and 14 on page 231

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Rational Zeros Theorem

• Let f be a polynomial function of degree 1 or
  higher of the form
• F(x) anxnan-1xn-1a1xa0
• an and ao not zero
• Where each coefficient is an integer. If p/q, in
  lowest terms, is a rational zero of f, then p
  must be a factor of a0 and q must be a factor of
  an.

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Possible Rational Zeros

• List the potential rational zeros of
• F(x) x5 - x4 2x2 3
• A0 is 3, so p is a factor of 3.
• P 1, 3
• An is 1 so q is a factor of 1.
• Q 1

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Possible Rational Zeros

• List the potential rational zeros of
• F(x) x5 - x4 2x2 3
• P 1, 3
• Q 1
• p/q 1, 3

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Finding the Real Zeros of a Polynomial

• Use the degree of the polynomial to determine the
  maximum number of zeros.
• If the polynomial has integer coefficients, use
  the Rational Zero Theorem to identify the
  rational numbers that could possibly be zeros.
• Use a graphing calculator to graph the polynomial.

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Finding the Real Zeros of a Polynomial

• Use the graph to help find the zeros from the
  possible list.
• If there was no graph, then use substitution to
  find the possible zeros that work.
• Create factors out of the zeros.
• Use synthetic division or long division to help
  factor the equation, or use other factoring
  techniques.

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Examples

• 22,30,40,42

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Solving Polynomial Equations

• Every polynomial function with real coefficients
  can be uniquely factored into a product of linear
  and/or irreducible quadratic factors.

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Solving Polynomial Equations

• A polynomial function, with real coefficients, of
  odd degree has at least one real zero.

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Use the Calculator to Find Zeros

• Place the equation in y1.
• Notice that if the graph starts above the x-axis,
  or has positive y values and then goes below the
  x-axis or has negative y values, then there is a
  zero or the graph will cross the x-axis as long
  as the function is continuous.
• Use the zero function on the calculator or the
  table to approximate the zeros.