End Behavior
How can we determine the end behavior of a polynomial algebraically (without using a graph)?
Leading term (degree and coefficient)
Elena is making an open-top box by cutting squares out of the corners of a piece of paper that is 10 inches wide and 15 inches long, and then folding up the sides. The width of the open-top box, in terms of x, will be represented by this expression.
10-2x
Factor x2-12x+36
(x-6)2
If polynomial P(x) is known to have a degree of 7, a zero at x = -5 with a multiplicity of 3, a zero at x = 1 with a multiplicity of 2, and a non-real zero, what is the final zero?
Non-real
Find the zeros of the function: f(x)=2x(x-4)(x-1)
Write the end behavior for a polynomial with a positive leading coefficient and an even degree.
x -> -oo, y -> oo
x -> oo, y -> oo
Elena is making an open-top box by cutting squares out of the corners of a piece of paper that is 12 inches wide and 14 inches long, and then folding up the sides. The volume of the open-top box, in terms of x, will be represented by this expression.
V(x)=(12-2x)(14-2x)(x)
Factor 2x2+6x-8
2(x-1)(x+4)
How can you determine the number of zeros in a polynomial?
By its degree (highest exponent)
List the zeros and their multiplicities from the polynomial.
y=x(x-3)^3(x+4)^5
x = 0 multiplicity 1
x = 3 multiplicity 3
x = -4 multiplicity 5
Describe the end behavior of the polynomial
g(x) = 4x5-3x3+7x
x -> -oo, y -> -oo
x -> oo, y -> oo
Elena is making an open-top box by cutting squares out of the corners of a piece of paper that is 10 inches wide and 15 inches long, and then folding up the sides. If the side lengths of her square cutouts are x inches, then what is a reasonable domain for this context?
0<x<5
Factor the expression x3-8
(x-2)(x2+2x+4)
How do you determine the number of non-real zeros of a polynomial?
Subtract the number of real zeros from the degree
Find the zeros of the function:
f(x) = -3(2x-1)(x+6)(3x+7)
Describe the end behavior of the polynomial
h(x) = 2x2-5x5+6x3-10
x -> -oo, y -> oo
x -> oo, y -> -oo
Elena is making an open-top box by cutting squares out of the corners of a piece of paper that is 10 inches wide and 15 inches long, and then folding up the sides. If the side lengths of her square cutouts are x inches, then write an equation to represent the volume of the box.
V(x)= x(10-2x)(15-2x)
Factor 2x3+1024
2(x+8)(x2-8x+64)
Identify the number of real and non-real zeros of 3x3-3
1 real, 2 non-real
Find the zeros of this function: y=2x3+14x2+12x
x = 0, -6, -1
Describe the End Behavior of f(x)=-4x4+3x5-x2+7
x -> -oo, y -> -oo
x -> oo, y -> oo
Elena is making an open-top box by cutting squares out of the corners of a piece of paper that is 10 inches wide and 15 inches long, and then folding up the sides. If the side lengths of her square cutouts are x inches, then what value of x will give her the greatest volume? What is the volume?
x = 1.96
V(x) = 132.03
Factor 3x3-648
3(x-6)(x2+6x+36)
What is the number and type of roots of x3+2x?
1 real, 2 non-real
The zeros of:
3x3 + 30x2 + 63x
x = 0, -3, -7