regents review- 4,5,6

Question 1

GCF

Answer 1

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)

Question 1

DPS

Answer 1

difference of perfect squares, follows form a2 -b2 = (a+b) (a-b) ex. 49x2-81y2 = (7+9y) (7x-9y)

Question 2

to factor a sum of perfect squares

Answer 2

a2 + b2 = (a+bi) (a-bi) ex. 49x2 + 81y2 = (7x+9yi) (7x-9yi)

Question 3

to factor a sum of perfect cubes

Answer 3

a3+b3= (a+b)(a2-ab+b2)
ex. x3+ 27= (x+3) (x2-3x+9)

Question 4

to factor a difference of cubes, use a3-b3

Answer 4

a3-b3 = (a-b) (a2+ab-b2) ex x3-8 –> (x-2) (x2+2x+4)

Question 5

to factor a trinomial with a=1

Answer 5

find 2 #s that multiply to c but add up to b, then use shortened(????)
ex. x2-9x+20= (x-4) (x-5)

Question 6

to factor a trinomial with a>1

Answer 6

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)

Question 7

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

Answer 7

ex. 3x3-16x2-12x+64= x2(3x-16) -4(3x-16) –> (x2-4)(3x-16) –> (x+2)(x-3)(3x-16)

Question 8

when factoring completely, always check for gcf then look for other methods

Answer 8

4 methods for solving a quadratic: factoring, completing the square, quadratic formula, graphing

Question 9

if a quadratic equation has complex roots, always write in simplest a+bi form

Question 10

to solve quadratic inequalities algebraically, change to = sign, solve, make a # line and check test

Question 11

expression for the sum of roots of a quadratic ax2+bx+c = -b/a
and product of the roots is c/a

Question 12

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)

Question 13

discriminant: b2-4ac

Question 14

if 2 different real roots

Answer 14

b2-4ac>0
- perfect square/not a perfect square
- 2 irrational, rational, unequal roots

Question 15

if 2 = roots, b2-4ac=0

Answer 15

b2-4ac=0
- 2 irrational, rational, equal roots (double root)

Question 16

if no real roots

Answer 16

b2-4ac <0
- 2 imaginary roots

Question 17

a quadratic can be written….

Answer 17

• standard form: y= ax2+bx+c
• vertex form: y= a(x-h) 2+ k
  vertex: (h,k)
  axis of symmetry: x=h

Question 18

to write a quadratic y=ax2+ bx+c in vertex form by completing the square

Answer 18

group x terms together, add (b/2)2 inside () but subtract it outside (), factor inside ()

Question 19

the graph of a quadratic y=ax2+bx+c is a parabola and has domain (-infinity, infinity)

Answer 19

if a is positive and (negative infinirt, nax] is a is negative

Question 20

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 ()

Question 21

a graph is increasing when it is rising (slope) and decreasing when - slope

Answer 21

always state intervals using x values from the graph and write with ()

Question 22

if question asks for max or min value –> solve for y
if question asks when a max or min occurs –> solve for x

Answer 22

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

Question 23

if q asks for the time at which the object hits the ground, set equation equal to 0

Answer 23

solve for x

Question 24

locus definition of a parabola

Answer 24

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

Question 25

circle equation: (x-h)2 + (y-k)2 = r2 where (h,k) is the center and r is the radius

Question 26

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

Question 27

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]

Question 28

to solve 3 equations in 3 variables algebraically use double elimination

Question 29

if system of equations has no solution, the system is inconsistent, if the system of equations has infinite solutions the system is dependent

Question 30

if a power function f(x) = ax to the hth power is an even exponent

Answer 30

positive and negative inputs have the same output

Question 31

if a power function f(x)= ax to the hth power is an odd exponent

Answer 31

positive and negative inputs have opposite outputs

Question 32

a polynomial is the sum and/or difference of power functions with whole number exponents

Answer 32

the end behavior of any polynomial is determined by its highest power term

Question 33

when solving higher order polynomial equations, the highest exponents tells you the number of solutions

Answer 33

if solving higher order polynomial equations graphically, be sure to show 4 parts of your work: y1 & y2, window, labeled sketch, solution

Question 34

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

Question 35

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

Question 36

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

Question 37

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)

Question 38

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

Question 39

after dividing f(x) by x-k, if the remainder is 0, then x-k is a factor of f(x)

Question 40

when setting up long division or synthetic division, don’t forget to use placeholder of 0 if a term is missing

Question 41

use synthetic division if dividing by a polynomial in the form x-c where c is the constant

Question 42

if a polynomial f(x) is divided by x-c, the remainder is f(c)

Answer 42

remainder theorem

Question 43

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

Question 44

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

Question 45

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

Question 46

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

Question 47

when subtracting rational expressions make sure to distribute the negative to all terms, don’t simplify or cancel until after adding/subtracting

Question 48

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

Question 49

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 {}

Question 50

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

Question 51

use 1/time alone + 1/time alone = 1/time together to solve work problems