any second degree (highest power of x is 2) polynomial equation 2x² + 5x = 12

x² + bx + c when c is positive, find 2 numbers with ... when c is negative, find 2 numbers with ...

the same sign as the middle term: x² - 5x + 6 = (x - 3)(x - 2) opposite signs whose sum is the coefficient of the middle term: x + 3x - 18 = (x + 6)(x - 3)

all terms have a variable raised to a positive integral power (no fraction or negative powers) just terms with a coefficient no logs, sines or cosines allowed ex: 4x⁵ - 6x³ + x² - 5x + 2

Chapter 1 - 4: Calculus PreReqs Flashcards by Alexandra Miranda

solving quadratic equations: completing the square

3x² = 24x + 27

put the variables on one side and the constant in the other: 3x² - 24x = 27
divides both sides by coefficient of x²: x² - 8x = 9
divide coefficient of x by 2 and square it, then add to both sides: x² - 8x + 16 = 9 + 16
factor left side (factor always has same # from prior step): (x-4)(x-4) = 25 → (x-4)² = 25
use square root property: x-4 = √25 → x-4 = ±5
solve: x-4 = ±5 → x = 9 or -1

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The quantity b2 - 4ac, which appears under the radical sign in the quadratic formula
determines the number and type of solutions
b2 - 4ac > 0: Two unequal real solutions
b2 - 4ac = 0: One solution (a repeated solution) that is a real number
b2 - 4ac < 0: No real solutions

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solving quadratic equations: quadratic formula

2x² - 6x + 1 = 12

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solving quadratic equations: factoring

2x² - 5x = 12

bring all terms to one side: 2x² - 5x - 12 = 0
factor: (2x + 3)(x -4) = 0
set each factor to 0 and solve: 2x + 3 = 0 → 2x = -3 → x = -3/2
set each factor to 0 and solve: x - 4 = 0 → x = 4
two solutions: -3/2, 4

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differences of two cubes

64x³ - 125

express each term as the cube of a monomial: 64x³ + 125 = 4x³ - 5³
convert 4x³ - 5³ to (a³ - b³) = (a - b)(a² + ab + b²)
(4x - 5)(4x² - 4x5 + 5²) → (4x - 5)(4x² - 20x + 25)

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factoring sum of cubes

x³ + 8

express each term as the cube of a monomial: x³ + 8 = x³ + 2³
convert x³ + 2³ to (a³ + b³) = (a + b)(a² - ab + b²)
(x + 2)(x² - x2 + 2²) → (x + 2)(x² - 2x + 4)

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quadratic equation

any second degree (highest power of x is 2) polynomial equation
2x² + 5x = 12

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x² + bx + c

when c is positive, find 2 numbers with …
when c is negative, find 2 numbers with …

the same sign as the middle term: x² - 5x + 6 = (x - 3)(x - 2)
opposite signs whose sum is the coefficient of the middle term: x + 3x - 18 = (x + 6)(x - 3)

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factoring trinomials steps

x² + 6x + 8

find the first terms whose product is x²: (x, inside) (x, outside)
find 2 last terms whose product is 8: 8, 1 or 4, 2 or -8, -1 or -4, -2
select the terms whose outside product and inside product is 6x: 4, 2
answer: (x, 4) (x, 2) (outside 2 ⋅ x + inside 4 ⋅ x = 6x)
x² + 6x + 8

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factoring by grouping

x³ + 4x² + 3x + 12

group terms w/common factor: (x³ + 4x²) + (3x + 12)
take out GCF: x²(x + 4) + 3(x + 4)
take out GCF: (x + 4)(x² + 3)

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factoring the greatest common factor

x²(x + 3) + 5(x + 3)

if you are adding, the answer is multiplication
GCF is (x + 3), so pull it out
(x + 3)(x² + 5)

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factoring the greatest common factor

18x³ + 27x²

9 and x² are the greatest integer and expression that divides in 18 & 27 and x² & x³
9x²(2x) + 9x²(3)
9x²(2x + 3)

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what is the degree of the following polynomial:

4x⁵ - 6x³ + x² - 5x + 2

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degree of a polynomial

the polynomial’s highest power of x

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polynomial

all terms have a variable raised to a positive integral power (no fraction or negative powers)
just terms with a coefficient
no logs, sines or cosines allowed
ex: 4x⁵ - 6x³ + x² - 5x + 2

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difference of cubes

(a³ - b³) = (a - b)(a² + ab + b²)

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sum of cubes

(a³ + b³) = (a + b)(a² - ab + b²)

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a difference of squares (a² - b²) _____ be factored, but a sum of squares (a² + b²) _____

can
can’t

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factor the following

(3x²)² - (5)²

(3x² - 5)(3x² + 5)

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rewrite the following to show a difference of squares (a² - b²)

9x⁴ - 25

(3x²)² - (5)²

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LogcC =

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Logc1 =

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rewrite log₃81 = x

3ˣ = 81

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logarithms

2³=8 as log is …

log₂8 = 3
log base 2 of 8 = 3

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logarithms rewrite log₂8 with base b

* log₂8 = log_b8 / log_b8

logarithms convert log₁₃4 to natural logs (base e)

* log₁₃4 = log_e4 / log_e13 * log_e4 / log_e13 = ln4 / ln13

if you have an even number root/index, you need the _____ value on the answer, because whether the root is positive or negative, the answer is \_\_\_\_\_. if it's an odd number root, you don't need the \_\_\_\_\_.

* absolute * positive * bars

rewrite

(x²)³=

x⁶

x^-4

x² • x³ =

x⁵

rewrite as a root

rewrite x^-3 =

1/x³

x⁰ =

* 1 * if x = 0, then 0⁰ = undefined

is cancelling allowed? a²b³(xy - pq)⁴(c + d) / ab⁴z(xy - pq)³ + 1

* no * + 1 breaks the chain of multiplication

is cancelling allowed? a²b³(xy - pq)⁴(c + d) / ab⁴z(xy - pq)³

* yes * a(xy - pq)(c + d) / bz

multiplication rule for cancelling

can cancel in a fraction only when it has an unbroken chain of multiplication through the entire numerator and denominator

algebraic expression

* anything w/out an equal sign * if it has an equal sign its an equation * behave like variables

dividing fractions

* flip the second fraction * multiply the numerators * multiply the denominators

multiplying fractions

* multiply the numerators * multiply the denominators

limits

* the microscope that zooms in on a curve * zooms in on a curve until its straight * curves represent changing quantities

convergent series

* you keep adding numbers * sum keeps growing * even though you add # forever and sum grows forever, sum of all infinitely is a finite number

divergent series

* series adds up to infinity * there are some exceptions

slope =

when you zoom in on curves they become \_\_\_\_\_

straight

integration

adding up small parts of something to get the total

differentiation

finding a derivative

derivative

* slope of a curve or steepness * simple rate (i.e. miles per hour)

**exponential** For any base where b \> 0 (b^x)^y =

b^xy

**exponential** For any base where b \> 0 1/ b^y =

b^-y

**exponential** For any base where b \> 0 b^x/ b^y =

b^x/ b^y = b^x-y

**exponential** For any base where b \> 0 b^xb^y =

b^xb^y = b^x+y

**logarithmic** For any base where b \> 0, b ≠ 1 log_b1 =

log_b1= 0

**logarithmic** For any base where b \> 0, b ≠ 1 log_bx =

log_bx= log_e x = ln x

**logarithmic** For any base where b \> 0, b ≠ 1 log_bb =

log_bb= 1

**logarithmic** For any base where b \> 0, b ≠ 1 log_bx^z =

log_bx^z= z log_bx

**logarithmic** For any base where b \> 0, b ≠ 1 log_b1/y =

log_b1/y = - log_by

**logarithmic** For any base where b \> 0, b ≠ 1 log_bx/y =

log_bx/y = log_bx - log_by

**logarithmic** For any base where b \> 0, b ≠ 1 log_bxy =

log_bxy = log_bx + log_by

**logarithmic** For any base where b \> 0, b ≠ 1 log_bb^x =

log_bb^x = x

**logarithmic** For any base where b \> 0, b ≠ 1 b^log_bx =

b^log_bx = x

Natural exponential functions

* b is a positive number not equal to 1 * Inverse of exponential functions * In the example, function passes through (0,1) and (1,e) * slope of e^x equals the value of e^x * when x = 0, e⁰ = 1 → value of e^x = 1, and slope = 1 * when x = 1, e¹ = e → the value of e^x = e, and slope = e * The area up to any x-value is also equal to ex * only exponential function with the property that the slope of the tangent line at x = 0 is 1, so e^x has both value and slope equal to 1 at x = 0 * Simplifies calculations * Inverse of natural logarithm function: ln(x) * ln(e^x) = x * They are the same curve with x-axis and y-axis flipped. * e(ln x) = x

Natural logarithm functions

* with base b = e * log_e(x) which is more commonly written ln(x) * Inverse of natural exponential function: e^x * e^{(ln x)} = x * They are the same curve with x-axis and y-axis flipped. * In the example, function passes through (0,1) and (e,1)

formula for area of triangle

(base • height) / 2

formula for area of square

width • height

**Natural logarithm** ln xy =

ln x + ln y

**Natural logarithm** ln (x/y) =

ln x - ln y

**Natural logarithm** ln |x| =

1 / x

**The number *e*** ln *e*^p =

**The number *e*** *e*^{x + y} =

*e*^x*e*^y

**The number *e*** *e*^{x - y} =

*e*^x / *e*^y

**The number *e*** *(e*^x)^p =

*ex*^y

**The number *e*** ln *(e*^x) =

**The number *e*** *b*^x =

**simplifying √ of exponents** when the exponent is even √x⁶

* divide exponent by 2 * write result outside of **√** * leave no variable inside √ √x⁶ * exponent = 6 * 6 / 2 = 3 * x³√ = x³ * x³ = x⁶

**simplifying √ of exponents** when the exponent is odd √x³

* subtract one from the exponent * divide exponent by 2 * write result outside of **√** * leave variable inside √ w/no exponent √x³ * exponent = 3 * 3 - 1 = 2 * 2 / 2 = 1 * 1x√ * 1x√x = x√x

**simplifying √ of exponents** √80x³y²

80x³y² * exponent = 3 * 3 - 1 = 2 * 2 / 2 = 1 x√80xy² * exponent = 2 * 2 / 2 = 1 xy√80x * 4 \* 4 \* 5 = 80 4xy√5x √80x³y^{2 =}4xy√5x

exponential growth & decay f(t) = A₀e^kt

* A₀ = original amount/size at time t = 0 * t = time elapsed * A = amount at time t * k * rate constant * represents growth rate * \> 0 = growth * \< 0 = decay

Equation for doubling time time required for a quantity (A₀e^kt) to double

Equation for half life