Real Zeros of Polynomial Functions

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Real Zeros of Polynomial Functions

• Written by Clinton Henry

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Table of Contents

• Long Division of Polynomials
• Synthetic Division
• The Remainder and Factor Theorems
• The Rational Zero Test
• The End

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Long Division of Polynomials

• Using long division of polynomials is just about
  like using long division in regular numbers. It
  s set up the same with one exception. The
  divisor is usually linear factor instead of a
  constant.
• Lets try one.

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Long Division of Polynomials (cont.)
multiply x-2 by 6x2 subtract 6x3-12x2 multiply
x-2 by -7x subtract -7x214 multiply x-2 by
2 subtract 2x-4
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Long Division of Polynomials (cont.)

• Since we have no remainder, we know x-2 is a
  factor. We also can use several other types of
  factoring to break down the answer to get factors
  of (x-2)(2x-1)(3x-2).
• This means we have x-intercepts when

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Long Division of Polynomials (cont.)

• Lets try to use long division on the following
• We notice that there is not enough terms to get a
  proper division so we will have to use
  placeholders like so.

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Long Division of Polynomials (cont.)
multiply x-1 by x2 subtract x3-1x2 multiply x-1
by x subtract x2-1 multiply x-2 by 1 subtract x-1
The result ends up with something that cannot be
divided that easily until further in this section.
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Synthetic Division

• When trying to use synthetic division, you must
  be dividing by a linear factor with a leading
  coefficient of 1.
• A linear factor of (x k) has a zero k.
• There is also two patterns to remember, vertical
  pattern is adding and diagonal pattern is
  multiplied by k.

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Synthetic Division (cont.)

• Here is what we should be looking to do.

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Synthetic Division (cont.)

• Example 1 Lets divide using synthetic
  division the following

Remember x-k means k is zero so x3x-(-3) so
k-3. We also do not have a x3 term so we will
need a place holder.
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Synthetic Division (cont.)
Notice we have a remainder of 1 so our actual
solution to this synthetic division is the
following.
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The Remainder and Factor Theorems

• The main idea of the Remainder Theorem is when a
  polynomial f(x) is divided by x-k, the remainder
  (r) is always equal to f(k).
• If we look at the last section on synthetic
  division, we ended up with a remainder of 1. We
  can also say f(-3)1.

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The Remainder and Factor Theorems (cont.)

• The Factor Theorem states that if we have a
  polynomial f(x) which we know has a factor of (x
  k) if and only if f(k)0 which means no
  remainder and must cross the x-axis at that
  moment.

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The Rational Zero Test

• The Rational Zero Test states that for every
  polynomial f(x) that has integer coefficients
  there every possible rational zero will be in the
  for of p divided by q.

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The Rational Zero Test (cont.)

• Lets do an example.
• Find all real zeros of the following.

The factors of the constant 12 are 1, 2, 3,
4, 6, 12. The factors of the leading
coefficient 10 are 1, 2, 5, 10.
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The Rational Zero Test (cont.)

• We can see we have 32 different possibilities.
  Looking at the original problem, we can probably
  try x2 because both the leading coefficient and
  constant are even. We might try a synthetic
  division to get the problem down to a quadratic.

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The Rational Zero Test (cont.)
We also can use the quadratic formula to see if
the other 2 zeros are real. (Any questions refer
to module 1).
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The End

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