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True or false? A polynomial function is a function involving only non-negative integer powers of x.
True
In order for a function to be a polynomial function, it must be _____ and ________.
Smooth and continuous
True or false? We can divide polynomials similar to how we divide numbers.
True.
According to the Rational Zeros Theorem, if every coefficient of a polynomial is an integer, then every rational zero of the polynomial has the form p/q, where p is a factor of the _____, and q is a factor of the ________.
constant; leading coefficient
What is the standard form of a quadratic equation?
f(x) = a(x - h)2 + k
Describe the end behavior of the polynomial:
P(x) = -2x4 + 5x3 + 4x - 7
[Hint: Consider the degree of the polynomial and the leading coefficient.]
y→-∞ as x→∞ , and y→-∞ as x→ -∞
Synthetic division is a quick method used when the divisor is of the form _____.
x - c
Find the rational zeros of P(x) = x3 - 3x + 2
1 and -2
What is the formula for the minimum or maximum of a quadratic function?
x = -b/(2a)
If P is a polynomial function and P(a) and P(b) have opposite signs, then there exists at least one value c between a and b for which P(c) = 0.
What is the name of this theorem?
Intermediate Value Theorem
Divide 2x3 -7x2 + 5 by x - 3 using synthetic division. Find the quotient and remainder.
Q(x) = 2x2 - x - 3
r = 4
Find all rational zeros of the polynomial, and then write the polynomial in factored form.
P(x) = x3 + 3x2 - 4
x = 1, -2
P(x) = (x - 1)(x +2)2
Express the function in standard form:
f(x) = 2x2 - 12x + 23
f(x) = 2(x - 3)2 + 5
What are the 4 steps for graphing polynomial functions?
1. Factor to find all the zeros.
2. Use test points to determine if the graph falls above or below the x-axis.
3. Determine the end behavior.
4. Graph.
Use long division to find the quotient and remainder.
P(x) = 8x4 + 6x2 - 3x + 1; D(x) = 2x2 - x + 2
Q(x) = 4x2 + 2x
r = -7x + 1
Write the equation in standard form and then find the minimum/maximum value.
f(x) = -x2 + x + 2
f(x) = -(x - 1/2)2 + 9/4
maximum = 9/4
Find the zeros of the function and then sketch its graph.
P(x) = x3 - 2x2 - 3x
x = -1, 0, 3
Let P(x) = x3 - 7x + 6. Show that P(1) = 0, and use this fact to factor P(x) completely.
[Hint: Use the Factor Theorem.]
P(1) = 13 - 7(1) + 6 = 0. That means that x - 1 is a factor.
Using synthetic division, we find that
P(x) = (x - 1)(x + 3)(x - 2)