Solving Polynomial Equations by Graphing

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Solving Polynomial Equations by Graphing
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Types of Equations

• Quadratic - has the form ax2 bx c 0
• Highest exponent is two (this is the degree)
• The most real solutions it has is two.

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Types of Equations

• Cubic - has the form ax3 bx2 cx d 0
• Highest exponent is three (this is the degree)
• The most real solutions it has is three.

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Types of Equations

• Quartic - has the form ax4 bx3 cx2 dx e
  0
• Highest exponent is four (this is the degree)
• The most real solutions it has is four.

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Types of Equations

• These keep on going up as the highest exponent
  increases.
• You dont need to know the names above quartic,
  but you do need to be able to give the degree.

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Solving Equations

• When we talk about solving these equations, we
  want to find the value of x when y 0.
• Instead of solve we call this finding zeros
  or roots.

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Solving Equations

• Get all the x or constant terms on one side.
• If you have a y or f(x), replace it with 0.

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Solving Equations

• The first way we are going to solve these
  equations is by graphing. (Yeah!!! More
  calculator stuff!!)
• Go to the graph menu on your calculator.

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Solving Equations

• Solve x2 - 4 y
• Replace y with 0.
• Plug in x2 - 4 into your calculator.
• Graph it and lets look at the graph.

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Solving Equations

• When we talk about the graph and we are looking
  for places where y 0, where will these points
  be?
• On the x-axis.
• So we are looking for the x-intercepts.

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Solving Equations

• So where does this graph cross the x-axis?
• (2, 0) and (-2, 0)
• If you cant tell from looking at the graph, go
  to F5 (gsolv) and then F1 (root).

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Solving Equations

• This should give you the first zero, to get to
  the second, hit the right arrow button.
• Note the zeros should be on the screen. If you
  cant see the x-intercepts, make your window
  bigger.

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Solving Equations

• So the solutions to this equation are x 2 or x
  -2.

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Solving Equations

• Find the solutions to
• f(x) x2 - 5x 6.
• x 3, 2
• Find the zeros of
• 0 x2 - 4x 4
• x 2

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Solving Equations

• How do we check our solutions?
• Plug in and see if the equation simplifies to 0.

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Solving Equations

• Lets look at quadratic equations for a minute.
• How many solutions should you look for?
• Two, one or zero.

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Solving Equations

• Lets look at some cubic equations.
• x3 - 1 0
• x 1
• x3 - 6x 1 f(x)
• Has three solutions.

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Solving Equations

• When we are solving cubic equations, we will have
  either 3, 2, or 1 real solution. You should
  never have no solutions.

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Solving Equations

• What about quartic equations?
• They look like W or M.
• They could have four, three, two, one, or no
  solution.

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Solving Equations

• Lets look at your graphing equations worksheet.

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Factoring

• For these last two methods for solving equations,
  we will be looking at only quadratic equations
  (degree 2).
• The next method we will look at is factoring.

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Factoring Quadratics

• We know that quadratic equations are set equal to
  0.
• We will factor the trinomial and set each factor
  equal to 0 to find our solutions.

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Factoring Quadratics

• x2 - 4 0
• Lets try the first graphing example and factor
  it.
• to factor x2 - 4 we use difference of squares.
• x2 - 4 (x - 2)(x 2) 0

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Factoring Quadratics

• Okay, lets take a side note for a second.
• If we multiply two numbers and get a product of
  0, what do the factors have to be?
• 3x 0, what does x have to be?

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Factoring Quadratics

• if ab 0, what do we know about a or b.
• Either a has to be 0, b has to be 0, or they both
  can be zero.
• This is the only way to get a product of 0.

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Factoring Quadratics

• Okay, back to factoring.
• (x - 2)(x 2) 0
• So x - 2 0, meaning x 2
• or x 2 0, meaning x -2
• So our solutions are x 2 and x -2.

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Factoring Quadratics

• Find the roots by factoring 2x2 8x - 24 0
• First, factor 2x2 8x - 24.
• 2(x 6)(x - 2).
• Set each factor (that contains an x) equal to
  zero.

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Factoring Quadratics

• x 6 0
• x -6
• x - 2 0
• x 2
• So x -6 or x 2.

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Quadratic Formula

• The last method we will use to solve quadratic
  equations is the quadratic formula.
• This is the only method that will ALWAYS work
  when trying to solve a quadratic equation.

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Quadratic Formula

• All the quadratic formula is is plugging in
  numbers.
• You dont need to worry about memorizing it.
  They give it to you on the SOL

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Quadratic Formula

• Lets look back the the general form of a
  quadratic equation.

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Quadratic Formula

• ax2 bx c 0
• a is the coefficient of the squared term.
• b is the coefficient of the x term.
• c is the constant.

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Quadratic Formula

• If one of these three terms doesnt exist, then
  the coefficient of that term will be ____?
• 0

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Quadratic Formula

• what is the quadratic formula?

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Quadratic Formula

• Lets look at an example.
• 3x2 - 4x 3 0
• a ?
• b ?
• c ?

• a 3
• b -4
• c 3

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Quadratic Formula

• Now lets plug it in.
• b -4, so -b -(-4) 4

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Quadratic Formula

• Simplify

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Quadratic Formula

• Keep going now.

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Quadratic Formula
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Quadratic Formula

• Find the zeros of r2 - 7r -18 0

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Quadratic Formula

• Find the zeros of r2 - 7r -18 0
• a 1
• b -7
• c -18

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Quadratic Formula

• Find the zeros of r2 - 7r -18 0

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Quadratic Formula

• Simplify

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Quadratic Formula

• Now lets examine our solution.
• We can break this into two equations.

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Quadratic Formula

• Now we can get our two solutions.

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Quadratic Formula

• Now you try some.
• pg. 357
• 10 - 13 (only solve them using the quadratic
  formula)