regents review- 4,5,6 Flashcards
GCF
greatest common factor, like distributive property in reverse ex. 6x2-10x = 2x(3x-5)
ex. x4-16 = (x2+4) (x2-4) –> (x2 +4) (x+2) (x-2)
DPS
difference of perfect squares, follows form a2 -b2 = (a+b) (a-b) ex. 49x2-81y2 = (7+9y) (7x-9y)
to factor a sum of perfect squares
a2 + b2 = (a+bi) (a-bi) ex. 49x2 + 81y2 = (7x+9yi) (7x-9yi)
to factor a sum of perfect cubes
a3+b3= (a+b)(a2-ab+b2)
ex. x3+ 27= (x+3) (x2-3x+9)
to factor a difference of cubes, use a3-b3
a3-b3 = (a-b) (a2+ab-b2) ex x3-8 –> (x-2) (x2+2x+4)
to factor a trinomial with a=1
find 2 #s that multiply to c but add up to b, then use shortened(????)
ex. x2-9x+20= (x-4) (x-5)
to factor a trinomial with a>1
find 2 #s that multiply to a(c) but add to b, then rewrite the trinomial with 4 terms and factor by grouping
ex. 3x3-16x2-12x+64 = x2 (3x-16) -4 (3x-16) –> (x2-4)(3x-16) –> (x+2) (x-1) (3x-16)
use factoring when you notice a polynomial with 4 terms and factor the GCF out of each pair (often times after factoring completely you will need to keep factoring, always factor completely
ex. 3x3-16x2-12x+64= x2(3x-16) -4(3x-16) –> (x2-4)(3x-16) –> (x+2)(x-3)(3x-16)
when factoring completely, always check for gcf then look for other methods
4 methods for solving a quadratic: factoring, completing the square, quadratic formula, graphing
if a quadratic equation has complex roots, always write in simplest a+bi form
to solve quadratic inequalities algebraically, change to = sign, solve, make a # line and check test
expression for the sum of roots of a quadratic ax2+bx+c = -b/a
and product of the roots is c/a
writing equation for a quadratic using the sum and the product to the roots: x2 -sumx +product =0 (if there’s denominators, multiply everything by the lcd to eliminate the denominators)
discriminant: b2-4ac
if 2 different real roots
b2-4ac>0
- perfect square/not a perfect square
- 2 irrational, rational, unequal roots
if 2 = roots, b2-4ac=0
b2-4ac=0
- 2 irrational, rational, equal roots (double root)
if no real roots
b2-4ac <0
- 2 imaginary roots
a quadratic can be written….
- standard form: y= ax2+bx+c
- vertex form: y= a(x-h) 2+ k
vertex: (h,k)
axis of symmetry: x=h
to write a quadratic y=ax2+ bx+c in vertex form by completing the square
group x terms together, add (b/2)2 inside () but subtract it outside (), factor inside ()
the graph of a quadratic y=ax2+bx+c is a parabola and has domain (-infinity, infinity)
if a is positive and (negative infinirt, nax] is a is negative
a graph is positive when it is above the x axis and negative when below x axis, always state these intervals using x values from the graph and always write the intervals with ()
a graph is increasing when it is rising (slope) and decreasing when - slope
always state intervals using x values from the graph and write with ()
if question asks for max or min value –> solve for y
if question asks when a max or min occurs –> solve for x
if a question asks for the time at which an object is at a given height, set the equation equal to the given height and solve for x (or t) and use calc
if q asks for the time at which the object hits the ground, set equation equal to 0
solve for x
locus definition of a parabola
states that a parabola equation in the form (x-h)2= 4p(y-k) where the vertex is (h,k) and p is the distance from v to f or v to d
circle equation: (x-h)2 + (y-k)2 = r2 where (h,k) is the center and r is the radius
if given a circle equation that is not in center radius form we complete the square twice (once for the xs and once for the ys) to get it into center radius form
- add (b/2)2 to both sides if there is a coefficient in front of the squared terms divide that term out first
to solve systems of 3 equations in 3 variables in a calc press 2nd x-1 (B) rref [A] to get solution press 2nd x-1 to edit a 3 x 4 to matrix [A]
to solve 3 equations in 3 variables algebraically use double elimination
if system of equations has no solution, the system is inconsistent, if the system of equations has infinite solutions the system is dependent
if a power function f(x) = ax to the hth power is an even exponent
positive and negative inputs have the same output
if a power function f(x)= ax to the hth power is an odd exponent
positive and negative inputs have opposite outputs
a polynomial is the sum and/or difference of power functions with whole number exponents
the end behavior of any polynomial is determined by its highest power term
when solving higher order polynomial equations, the highest exponents tells you the number of solutions
if solving higher order polynomial equations graphically, be sure to show 4 parts of your work: y1 & y2, window, labeled sketch, solution
factor theorem states that if r is a root of polynomial P(x), then x-r will be a factor of P(x) and if x-r is a factor, then r is a root
if a polynomial has a double root, it has a multiplicity of 2, if a root appears 3 times, it has a multiplicity of 3, etc etc
if given a specific point on the polynomial you can write a particular equation, be sure to include a, the leading coefficient, and plug the point in for (x, y) and solve for a
complex roots always come in conjugate pairs, so if given only one complex root don’t forget the second (x-1-2i) (x-1+2i)
after dividing f(x) by x-k, f(x) can be rewritten as f(x)= (x-k)q(x) + remainder where q(x) is the quotient
after dividing f(x) by x-k, if the remainder is 0, then x-k is a factor of f(x)
when setting up long division or synthetic division, don’t forget to use placeholder of 0 if a term is missing
use synthetic division if dividing by a polynomial in the form x-c where c is the constant
if a polynomial f(x) is divided by x-c, the remainder is f(c)
remainder theorem
to greaph polynomials from the roots, a root with multiplicity of 1 crosses x axis a root with multiplicity 2 touches the x axis and turns around a root with multiplicity 3 flattens out as it crosses the x axis
to prove a polynomial identity your goal is to simplify one side until it matches the other, do not move terms between the left and right sides
to multiply rational expressions, simplify numerators and denominators by factoring and canceling, then multiply top(top) and bottom(bottom) then use multiplication and make sure to simplify fully
to add/subtract rational expressions, first rewrite the expressions with a common denominator, then add or subtract the numerators and keep the denominator, simply if possible
when subtracting rational expressions make sure to distribute the negative to all terms, don’t simplify or cancel until after adding/subtracting
to simplify complex fractions, multiply all terms by the lcd of the denominators in the top and bottom of the whole fraction, the resulting fraction should no longer be a complex fraction, then factor/cancel and simplify the resulting fraction
to solve rational (fractional) equations, multiply each term by the lcd to clear out all of the denominators then solve the remaining questions and check your answer (make sure to reject any solution that gives 0 in the denominator) btw it’s also possible to have no solution or {}
to solve rational inequalities, first write the inequality as an equation and solve, then determine any critical values (values that make the denominator =0) mark the solutions and critical values on a number line and then check test points in each interval on the number line and determine the solution set
use 1/time alone + 1/time alone = 1/time together to solve work problems
