Download How do you decide whether a function is a polynomial function and

CH 4 – POLYNOMIAL FUNCTIONS PORTFOLIO PAGE
How do you decide whether a function is a polynomial function and what determines its
classification? How do you multiply “special products” of polynomials? (Lesson 4.1-1 & 2)
Degree – when the polynomial is in standard form, the degree is the highest power (exponent) on any variable term
DEGREE
CLASSIFIC
-ATION
0
constant
1
linear
2
quadratic
3
cubic
4
quartic
5
quintic
Example: Write in standard form, then state its degree, type, and
leading coefficient:
(x – 3)(x2 – 5x – 9)
Special Product Patterns
(a+b)(a-b) =
(a + b)2 =
(a + b)3 =
(a – b)2 =
(a – b)3 =
What information can you determine about the graph of a polynomial function by working
only with its equation? (Lesson 4.1-3)
A polynomial of degree n has at most ______ turns.
A polynomial of degree n has at most ______ x-intercepts.
If the leading coefficient of the polynomial is positive, the graph will _____________ to the right.
If the leading coefficient of the polynomial is negative, the graph will ____________ to the right.
If the degree of the polynomial is even, the graph will have the ____________ behaviors to the left & right.
If the degree of the polynomial is odd, the graph will have _____________ behaviors to the left & right.
A relative minimum or maximum of a function is the y-value of a point on the graph that is _____________/
_______________ than the nearby points on the graph.
Relative Min/Max on Graphing Calculator:
2nd Calc/Minimum(Maximum)/Left bound, Right bound, Guess.
What information can you determine about the graph of a polynomial function by working
only with its equation? (Lesson 4.1-4)
FACTOR THEOREM
THE EXPRESSION (X – a) IS A LINEAR FACTOR OF
A POLYNOMIAL IF AND ONLY IF THE VALUE a IS A
ZERO OF THE RELATED POLYNOMIAL FUNCTION
If x – 3 is a factor of the polynomial, then 3 is
the zero.
If 3 is a zero of the polynomial, then x – 3 is a
factor.
Example: Given the following zeros, write the polynomial function in
standard form:
1) 0, 3 (multiplicity 2)
M. MURRAY
************
Zeros with EVEN multiplicity
will ____________ at x-axis
Zeros with ODD multiplicity will
____________ at x-axis
EQUIVALENT STATEMENTS ABOUT POLYNOMIALS
-4 is a solution of the equation x2 + 3x – 4 = 0
-4 is an x-intercept of the graph y = x2 + 3x – 4
-4 is a zero of the function y = x2 + 3x – 4
x + 4 is a factor of the expression x2 + 3x – 4.
What information can you determine about the graph of a polynomial function by working
only with its equation? (Lesson 4.1-4)
Exs. Given the following zeros, write the polynomial
function in standard form:
1) y = x(x + 2)2
degree________
Interval of increase:
2) y = -(x + 3)2(x – 4)2 degree______
Interval of decrease:
Where is y > 0?
Where is y < 0?
Relative minimum:
Relative maximum:
How is division used in evaluating and solving polynomials? (Lesson 4.3 Dividing)
LONG DIVISION: To divide polynomials using long
division, follow the long division algorithm. Write
the result in fractional form.
Example:
1) Divide 9x3 – 48x2 + 13x + 3 by x - 5
SYNTHETIC DIVISION: Synthetic division can be
used only when the divisor is of the form x - a
Example:
2) Divide 6x3 – 4x2 + 14x – 8 by x + 2
Example: Use synthetic division and the Remainder
Theorem to evaluate P(x) = x3 – 4x2 + 3x – 2 for P(2)
REMAINDER THEOREM
IF A POLYNOMIAL OF DEGREE 1 OR GREATER IS DIVIDED BY
X – a, THE REMAINDER IS P(a).
(THIS MEANS THE REMAINDER IS EQUAL TO THE VALUE OF
THE POLYNOMIAL WHEN a IS SUBSTITUTED IN FOR X)
How do you choose the best method for solving a polynomial equation?
(Lesson 4.4 -- Factoring)
3 Types of Factoring:
1) Sum/Difference of Cubes
2) Grouping
3) Polynomials in Quadratic form
Factor:
1) 8x3 + 1 = 0
2) x 4 + 3x2 – 28 = 0
Sum and Difference of Cubes
a3 + b3 = (a + b)(a2 – ab + b2)
*signs: +, - , +
a3 – b3 = (a – b)(a2 + ab + b2)
*signs: -, +, +
3) 2x 3 – 8x2 + 3x – 12 = 0
M. MURRAY