End Behavior
What is the end behavior of f(x) = -3x^4 + 5x^2 - 1 as x \to \infty and x \to -\infty ?
As x \to \infty, y \to -\infty and as x \to -\infty, y \to -\infty (both ends point down).
What are the zeros of P(x) = (x - 4)(x + 2)(2x - 3) ?
x = 4, x = -2, and x = \frac{3}{2}
Expand (x + 4)(x - 4) .
x^2 - 16
What is the y-intercept of f(x) = x^3 - 4x^2 + 7 ?
(0,7)
If P(2) = 0, what do we know is a factor of P(x) ?
(x-2)
Which term of the polynomial P(x) = 4x - 2x^3 + 7x^5 - 12 determines its end behavior as x gets very large?
The leading term, 7x^5 .
Does the graph of f(x) = (x - 2)^2 cross or bounce off the x-axis at x = 2?
Bounce (it touches and turns around because the exponent is 2).
Multiply x(x^2 - 3x + 2) .
x^3 - 3x^2 + 2x
What shape is the graph for a degree 4 polynomial?
When setting up long division for (x^3 - 7x + 2) \div (x - 1), what term must you insert as a placeholder?
0x^2
Describe the end behavior of an odd-degree polynomial with a negative leading coefficient.
As x \to \infty, y \to -\infty and as x \to -\infty, y \to \infty (starts up on the left, goes down on the right).
Write a factored polynomial with zeros at x = 0, 1, and -4.
f(x) = x(x - 1)(x + 4)
In an area diagram for (x + 3)(x + 5) , what are the two terms in the middle diagonal that combine?
3x and 5x (combining to 8x)
True or False: A degree 3 polynomial can cross the x-axis up to 3 times.
True
x^2-x-6\div(x-3)=
(x+2)
For g(x) = 0.5(x - 2)^2(x + 1)(x - 4) , what is the limit as x \to -\infty ?
g(x) \to \infty (Degree is 4, leading coefficient is positive 0.5).
What are the two zeros of g(x) = x^2(x - 6) and their multiplicities?
x = 0 (multiplicity 2) and x = 6 (multiplicity 1).
Multiply (x - 2)(x^2 + 2x + 4) .
x^3 - 8
If a box volume function is V(x) = x(10 - 2x)(8 - 2x), what is the maximum possible value for x before a side length becomes 0?
x = 4 (from 8 - 2x = 0 ).
Find the intersection point of f(x) = x^2 and g(x) = 2x.
(0,0) and (2,4)
Write a possible equation for a polynomial whose degree is 3, leading coefficient is negative, and has a y-intercept at (0, 6).
f(x) = -x^3 + 6 or f(x) = -(x - 1)(x + 1)(x - 6)
Find the y-intercept of P(x) = (x + 2)(x - 1)(x - 3) .
(0, 6) (Multiply the constants: (2)(-1)(-3) = 6).
If an area diagram yields x^3 + 2x^2 + 3x^2 + 6x , what factored expression was being multiplied?
(x + 2)(x^2 + 3x) or x(x + 2)(x + 3)
A graph goes up from the left, reaches a peak at (1, 4), and goes down. On what interval is it increasing?
For x < 1 (or (-\infty, 1)).
Is (x - 1) a factor of P(x) = x^3 - 4x + 3 .
Yes. No remainder!