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- Dilation by a constant
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- This pages
- Definition & Notation
- Polynomial Computation
- Possible Roots
- Quadratic, Cubic, Quartic Formulas
- Complex Roots
- Simple definition: polynomial
- Precalc definition
- -- an algebraic expression with variable x,
- a
_{n}x^{n}+ a_{n-1}x^{n-1}+ a_{n-2}x^{n-2}+ ... +a_{2}x^{2}+ a_{1}x^{1}+ a_{0}x^{0}, - in which the
constants
a
_{n}, a_{n-1}, a_{n-2}, ... , a_{2}, a_{1}, a_{0}are real, - and serve as coefficients or the constant term,
- and in which all exponent of x are counting numbers.
- The polynomial is named or identitied by the term of the highest power.
- constant term: a
_{0}x^{0}or a_{0} -
linear term: a
_{1}x^{1} -
quadratic term: a
_{2}x^{2} - cubic term: a
_{3}x^{3}. - quartic term: a
_{4}x^{4} - quintic term: a
_{5}x^{5} - Polynomials of even powers are similiar as they
- "start high & end high."
- Polynomials of odd powers are similiar as they
- "start low & end high."
- See highest degree for the impact on the graph and features of the polynomial.
- Leading coefficient --
- The coefficient of the term of highest degree.
- For polynomial a
_{n}x^{n}+ ..., - If a
_{n}is positive, the polynomial is not reflected. - If a
_{n}is negative, the polynomial is reflected. - See leading coefficient and the impact on the graph and features of the polynomial including overall and end behavior.
- All rules of arithmetic and algebra hold for polynomials.
- You may add, subtract, multiply, divide, and raise a polynomial to a power.
- All rules of the computation of functions hold for polynomials.
- Since a polynomial is a function, you may add, subtract, multiply, divide, and create compositions of polynomials.
- Composition (creating from other) of functions
- sum (f + g)(x)
- difference (f - g)(x)
- product (f · g)(x)
- quotient (f ÷ g)(x)
- composition (f o g)(x)= f(g(x))
- Play with Polynomial Graphs polynomial spreadsheet
- -- displays polynomial graphs in linear & quadratic factor
form and a
_{n}x^{n}+ ... form - Division Algorithm - Use division to rewrite a function as the
- product of two factors plus a remainder.
- f(x) = (quotient function) · (divisor function) + (remainder function)
- If the remainder is 0, the divisor evenly divides the original function.
- Examples of the above.
- ex 1. y = x
^{2}+ 4x + 4 = (x + 2)(x + 2) + 0 - ex 2. y = x
^{2}+ 4x + 5 = (x + 2)(x + 2) + 1 - ex 3. y = x
^{2}+ 6x + 4 = (x + 2)(x + 2) + 2x - ex 4. y = 2x
^{2}+ 4x + 4 = (x + 2)(x + 2) + x² - Factor Theorem --
- The polynomial (x-r) is a divisor (and factor) of the polynomial f(x) if and only if
- r is a root or zero of f(x)
- Examples of the above.
- ex 1. (x
^{2}+ 4x + 4)/(x + 2) = (x + 2) + 0, - because x = -2 is a root of y=x
^{2}+ 4x + 4 - ex 2.(x
^{2}+ 4x + 5) = (x + 2)(x + 2) + 1 - because x = -2 is not a root of y=x
^{2}+ 4x + 5. - When x = -2, y=x
^{2}+ 4x + 5 is equal to 1. - Integral Zero Theorem --
- The integer r is a root of the polynomial f(x) if and only if
- r is a
divisor of the constant term, a
_{0}. - Examples of the above.
- ex 1. -2 is a root of y=x
^{2}+ 4x + 4, - and -2 is a factor of 4, the constant term.
- Rational Zero Theorem --
- The rational number, fraction, p/q is a root of the polynomial f(x) if and only if
- p is a factor of the constant term and
- q is a factor of the leading coefficient.
- Examples.

Solve: Solution(s): p's, factors of

constant termq's, factors of

leading coefficientp/q's, possible

rational root

3x + 4 = 0 x = -4/3 ± 1, ±2, ±4 ±1, ±3 ± 1/1, ±2/1, ±4/1, ± 1/3, ±2/3, ±4/3

4x - 3 = 0 x = -3/4 ± 1, ±3 ±1, ±2, ±4 ± 1/1, ±3/1, ± 1/2, ±3/2, ± 1/4, ±3/4

x + 3 = 0 x = -3 ± 1, ±3 ±1 ± 1/1, ±3/1

(x + 2)(x - 5) = 0

x^{2}- 3x - 10 = 0x = -2,

x= 5± 1, ±2, ±5 ±1 ± 1, ±2, ±5

x ^{3}+ 5x^{2}+ 2x - 8 = 0x = ? ± 1, ±2, ± 4, ±8 ±1 ± 1, ±2, ± 4, ±8

- Irrational Zeros --
- Use the quadratic formula for 2nd degree polynomials.
- Use the The "Cubic Formula" by Helmut Knaust at http://www.sosmath.com/algebra/factor/fac11/fac11.html for 3rd degree polynomials.
- Use the "Quartic Formula" by planetmath.org at http://planetmath.org/encyclopedia/QuarticFormula.html for 4th degree polynomials.
- There DO NOT EXIST formulas for 5th degree or higher polynomials.
- Approximate roots by determining sign changes in successive values of the function.
- Complex Roots
- Complex roots come in pairs.
- If a + b
*i*is a root, then its conjugate, a - b*i*is a root. - Seek these using the quadratic formula.
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