POLYNOMIALS

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POLYNOMIALS

• CHAPTER 9
• Sections 9.1 to 9.5

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ADDING, SUBTRACTING, MULTIPLYING POLYNOMIALS

• Sections 9.1 9.2

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A polynomial is..

• a term or a sum of terms.
• Each term is the product of a real-number
  coefficient and a variable with a whole-number
  exponent.

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The degree of a polynomial is the greatest
exponent. This polynomial has degree 3.
The exponent of this variable is 1, since x1 x.
This is the linear term.

• 2x3 5x2 4x 6

This term is called the constant term of the
polynomial. Note that 6 is the same as 6x0, since
x0 1.
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2x3 5x2 4x 6

• A polynomial whose exponents decrease from left
  to right is said to be in standard form.

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Example 1

• Tell whether each expression is a polynomial. If
  so, write the polynomial in standard form and
  state its degree. If not, explain why not.
• a) x2 8x3 2 3x4 7x

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Example 1

• Tell whether each expression is a polynomial. If
  so, write the polynomial in standard form and
  state its degree. If not, explain why not.
• b) 4x -3 9x x1/2 10

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Example 1

• Tell whether each expression is a polynomial. If
  so, write the polynomial in standard form and
  state its degree. If not, explain why not.
• c) 7r4 2r3 5r2

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Example 1

• Tell whether each expression is a polynomial. If
  so, write the polynomial in standard form and
  state its degree. If not, explain why not.
• d) 6.4m m3 pm5 m2

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Example 2

• Simplify
• (3x4 8x3 x2 2x 6) (9x4 4x2 2x 1)

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Example 3

• Simplify
• (7x3 5x2 x 4) ( 3x2 5x 7 6x3)

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Example 4

• Simplify
• (2x 3)(4x2 5x 8)

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Example 5

• Simplify
• ( 3x2 x 4)(7x2 6x 5)

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Assignment

• p 393 394 2 8 even, 20 30 even
• p 401 2 12 even

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DIVIDING POLYNOMIALS

• Section 9.2

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Remember
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Example 1
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Example 2

• (y3 4y2 20y 1) (y 3)

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Example 3
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Example 4
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You do
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What about?

• (4x4 4x3 11x2 16x 5) (2x2 x 5)

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Assignment

• p 403 21 29 all

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SOLVING CUBIC EQUATIONS

• Section 9.4

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A cubic function is

• a polynomial function of degree 3.
• Standard form f(x) ax3 bx2 cx d
• Intercept form f(x) a(x p)(x q)(x r)

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Example 1

• Let f(x) 2(x 1)(x 3)(x 4). Find the
  x-intercepts of the graph of f.

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Remember

• phrases that mean the same thing
• Roots of a function
• Solutions of the function
• Zeroes of the function
• x-intercepts of the function

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Example 2

• Find an equation for the cubic function g whose
  graph is shown.

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Synthetic Substitution

• Evaluate 2x3 x2 3x 7 when x 20

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Synthetic Substitution

• Evaluate 5x3 4x2 x 2 when x -3

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Synthetic Substitution

• Evaluate 6x3 x2 1 when x 4

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FACTOR THEOREM

• Let f(x) be a polynomial.
• Then x k is a factor of f(x) if and only if
  f(k) 0

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Finding Integral Zeroes of Polynomial Functions

• Let f be a polynomial function with integral
  coefficients. Then the only possible integral
  zeroes of f are the divisors of the constant
  term.
• For example, f(x) 3x3 x2 19x 10. The
  possible integral zeroes of f are the divisors of
  10 1, 2, 5, and 10.

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Use synthetic substitution to show that f(x)
3x3 x2 19x 10 has only one integral zero.
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Example 4

• Find the zeroes of f(x) x3 4x2 5x 8.

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Example 5

• Find the zeroes of f(x) 10x3 33x2 23x 6

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Assignment

• p 416 417 1 9 odd, 13 19 odd, 23, 24, 26

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Finding the Zeroes of Polynomials

• Section 9.5

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Real Zeroes of Polynomial Functions

• A polynomial function of degree n has at most n
  real zeroes.

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For example a cubic function can have 1, 2, or 3
real zeroes.
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This function has 1 real zero

• because it has a
• triple root at (2, 0)

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This function has 2 real zeroes

• because it has a
• double root
• at (1, 0)

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It is possible that not all of the zeroes are
real numbers

• This function has only
• 1 real zero and the
• other two are imaginary.

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Example 1

• Find the zeroes of f(x) x3 x2 7x 15.

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Fundamental Theorem of Algebra

• A polynomial function of degree n has exactly n
  complex zeroes, provided each double zero is
  counted as 2 zeroes, each triple zero is counted
  as 3 zeroes, and so on.

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Rational Zeroes Theorem

• Let f be a polynomial function with integral
  coefficients. Then the only possible rational
  zeroes of f are p/q where p is a divisor of the
  constant term of f(x) and q is a divisor of the
  leading coefficient.

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In other words

• Possible rational zeroes of a polynomial function
  are

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Example 2

• Find the possible rational zeroes of
• f(x) 4x3 7x2 2x 3

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Example 3

• Find the zeroes of
• f(x) 9x4 6x3 13x2 12x 2

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Assignment

• p 423 424 9 17 odd

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Finding the Zeroes of Polynomials (cont.)

• Section 9.5

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Assignment

• p 423 424 10 18 even

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Exploring Graphs of Polynomial Functions

• Section 9.3

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Assignment

• Worksheet

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Summary Assignment (Review)

• p 423 2, 4
• p 458 459 1, 3 14