Ch. 9 & 10 Flashcards
polynomial
an expression that can be written as a monomial/sum of monomials w/ whole # exponents
degree
(of a polynomial) the largest exponent
what can a polynomial NOT be?
it can’t have negative or fractional exponents it can’t have variables in denominators
polynomial equation
an equation that can be written w/ a polynomial as 1 side & 0 as the other side
standard form
(of polynomials) from left to right, exponents go from largest to smallest
rational number
can be written as quotient of 2 integers
rational expression
an expression that can be written as the quotient of 2 polynomials
rational equation
an equation with only rational expressions on both sides
Product of Powers Rule
am • an = am + n
Quotient of Powers Rule
am/an = am - n
a does not equal 0
how to factor polynomials
- facor out any GCF’s
- factor normally (ax2 + bx + c)
ratio of the volume of two spheres
is equal to the ratio of the cubes of their diameters
Power of a Power Rule
(am)n = amn
Power of Product Rule
(ab)n = anbn
Power of a Quotient Rule
(a/b)n = an/bn
b does not equal 0
how to solve rational equations
- find LCD of all 3 & make the denominators all the same and cancel them out OR find LCD of left side denominators, make them the same, and then cross multiply
- simplify
extraneous solution
when a solution is not a solution of the original equation
(test it)
point of no return
the point where it would take as much time to return to the starting point as to continue on to the destination.
use t = d/r
d/r = d/r
cubic function
a polynomial function of degree 3
zero of a function
value of x that makes y = 0, same as x-intercept
double zero
cubic function w/ a squared factor
triple zero
a cubic function w/ a cubed factor
y = (x - 2)3
intersects x-axis once
y = (x - 2)2(x + 1)
intersects x-axis twice
y = (x - 2)(x + 1)(x + 4)
intersects x-axis three times
steps to solve a cubic equation w/out graphing
- put y on left side and set it to 0
- find the GCF of the right side if possible and factor it out
- if ax2 + bx + c format, use quadratic formula
how to solve cubic equations w/ calculator
- input equation
- find x-intercept (where y = 0)
square root of negative number
becomes i • pos. square root
parametric equation
equations where two variables are expressed in terms of a 3rd variable
parameter
3rd variable of a parametric equation
how to graph a parametric equation
- make a table of values (time, x, y)
- graph using x & y
OR
- change to PAR mode
- input equations
how to write an equation for y interms of x
- solve for t with either of the equations (preferably x)
- substitute into the other equation not used in step 1, t
- simplify
3D coordinate plane
how to find a coordinate on a coordinate plane
3D midpoint formula
(x1 + x2/2, y1 + y2/2, z1 + z2/2)
length of a diagonal of a rectangular prism
with edges of length x, y, and z & a diagonal of length d
d = square root of x2 + y2 + z2
3D distance formula
d = square root of (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2
equatons of circles and spheres centered at the origin
circle ⇒ x2 + y2 = r2
how to graph circle x2 + y2 = 36
- since it’s this form, center is (0,0) and r2 = 36; r = 6; sketch circle with center at Origin and (0,6) at top, (6,0) at right, (0, -6) at bottom, and (-6,0) at left
- use a graphing calculator; graph two semicricles ⇒ x2 + y2 = 36 ⇒ y2 = 36 - x2⇒ y = pos/neg square root of 36 - x2 to make circle circular, use Zoom Square
how to find the equation of a circle/sphere
- find center
- decide if it is at O or other and use the right equation
- find length of radius
- plug in radius
equations of circles not centered at origin
center (h,k) radius r
(x - h)2 + (y - k)2 = r2
