Polynomial
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Basics
End Behavior
& Multiplicity
& Multiplicity
Dividing
Polynomials
Polynomials
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Characteristics
Transformations & Systems
100
Identify the degree and leading term of the following polynomials:
1) f(x) = 6x2 + 8x3 - 5x4 - 3x + 23
2) g(x) = 2(x + 5)(x - 7)2
1) Degree = 4, LT = -5x4
2) Degree = 3, LT = 2x3
100
Determine the end behavior of the following function. Use proper notation.
f(x) = 4x3 - 8x2 + 3x4 + x - 10
`As ` x ->-\infty, f(x) ->\infty
`As ` x ->\infty, f(x) ->\infty
100
We know that x = -4 is a root of g(x) = x3 - 5x2 - 22x + 56. Use this information to write g(x) in factored form.
g(x) = (x + 4)(x2 - 9x + 14)
g(x) = (x + 4)(x - 2)(x - 7)
100
Identify the x & y intercepts
X-int: (-4, 0), (0, 0), (1, 0), (4, 0)
Y-int: (0, 0)
100
What would be the roots of g(x) = f(x - 3)?
(-1, 0) & (3, 0) & (4, 0) & (7, 0)
200
We are given the following polynomials:
f(x) = 4x2 - 5x3 + x - 7
g(x) = 7x4 + 2x3 - 8x - 1
Find t(x) = f(x) - g(x) and write you answer in standard form.
t(x) = (-5x3 + 4x2 + x - 7) - (7x4 + 2x3 - 8x - 1)
t(x) = -5x3 + 4x2 + x - 7 - 7x4 - 2x3 + 8x + 1
t(x) = - 7x4 - 7x3 + 4x2 + 9x - 6
200
Determine the end behavior of the following function. Use proper notation.
k(x) = -3x(x + 5)2(x - 1)3(x + 10)
LT = -3x7
`As ` x ->-\infty, k(x) ->\infty
`As ` x ->\infty, k(x) ->-\infty
200
What would be the remainder when this function is divided by the factor (x - 2)?
The remainder would be 3 because f(2) = 3.
200
Determine the relative extrema.
Rel max: (-2, 2.9) & (3, 6)
Rel min: (-4, 0) & (0.5, -0.3)
200
Find the point(s) where the functions below intersect.
f(x) = x^3+2x^2+x +7
g(x) = x^3 + x^2+3x+10
x^3+2x^2+x +7 = x^3+x^2+3x+10
x^2-2x-3 = 0
(x-3)(x+1) = 0
x =3, x=-1
f(3)=(3)^3+2(3)^2+(3) + 7
f(3) = 27+18+3+7=55
f(-1)=(-1)^3+2(-1)^2+(-1)+7
f(-1)=-1+2-1+7=7
Solutions: (3, 55) & (-1, 7)
300
Rewrite h(x) = (4x - 3)(7x2 - 2x4 + 3x - 8) in standard form.
h(x) = -8x5 + 6x4 + 28x3 - 9x2 - 41x + 24
300
Write the equation of g(x) graphed below.
g(x)=a(x+2)^3(x-1)^2(x-3)
-6=a(0+2)^3(0-1)^2(0-3)
-6=a(8)(1)(-3)
-6=-24a
a = 1/4
g(x)=\frac{1}{4}(x+2)^3(x-1)^2(x-3)
300
What is the remainder when the function below is divided by (x + 3)?
f(x)=x^3-2x^2+7x+45
The remainder is -21.
300
Determine the absolute extrema and the domain & range of the function below.
Abs Max: (4, 18)
Abs Min: N/A
Domain:
(-\infty, \infty)
Range:
(-\infty, 18]
300
What would be the y-intercept of k(x) = 5f(x + 2) - 4?
k(0) = 5f(0 + 2) - 4
k(0) = 5f(2) - 4
k(0) = 5(3) - 4
k(0) = 11
y-intercept: (0, 11)
400
We are given that f(x) = 3x2 - x + 4, g(x) = 2x + 3, and h(x) = 8x3 + 5x - 7. Determine the standard form of the equation r(x) = 2h(x) - f(x)g(x)
r(x)=2(8x^3+5x-7)-(3x^2-x+4)(2x+3)
r(x)=(16x^3+10x-14)-(6x^3+7x^2+5x+12)
r(x) = 10x^3-7x^2+5x-26
400
Sketch the graph of the following function. Include all x and y intercepts.
f(x)=-frac{1}{12}(x-4)^2(x+1)^3(x-2)(x+3)
400
The function given below has a relative maximum at (4, 0). Find the equation of f(x) in factored form.
f(x)=x^4-19x^3+128x^2-368x+384
Rel. max at (4, 0) means that (x - 4) is a factor with a multiplicity of 2 (function degree is 4).
f(x)= (x - 4)2(x2 - 11x + 24)
f(x) = (x - 4)2(x - 8)(x - 3)
400
Where is the function positive and negative?
`Pos: `(-\infty,-4)` U `(-4, 0)` U `(1,4)
`Neg: `(0, 1)` U `(4,\infty)
400
Determine the solution(s) to the system of equations given below.
f(x) = 2x+5
g(x) = x^3+x+5
2x+5 = x^3+x+5
0 = x^3-x
0=x(x^2-1)
0=x(x+1)(x-1)
x = 0, x = -1, x = 1
f(0) = 2(0)+5 = 5
f(-1) = 2(-1)+5 = 3
f(1) = 2(1)+5 = 7
Solutions: (0, 5) & (-1, 3) & 1, 7)