P(x) = anxn + an-1xn-1 + an-2xn-2 + ....+ a1x + a0 = 0 What are the sum and product of the roots?

Sum of the roots = - an-1/an Product of the roots = [a0/an](-1)n Where n is the degree of the polynomial

What does the discriminant D tell us for: 1. D \> 0 2. D = 0 3. D \<0

1. Roots are real and unequal 2. Roots are real and equal 3. Roots are imaginary (no real roots)

SAT 2 Subject test Flashcards by Christina Meyer

If polynomial P(x) is divided by x-a, what is the remainder?

P(a)

Because P(x)=(x-a)Q(x) + R

at x=a we have P(a)=R

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If f(a) = 0 then f(x) has a factor of (x-a)

Polynomial f(x) with a factor of (x-a) can be expressed as

f(x) = (x-a)Q(x)

Therefore, f(a) = 0 means that the remainder is 0

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P(x) = a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + ….+ a₁x + a₀ = 0

What are the sum and product of the roots?

Sum of the roots = - a_n-1/a_n

Product of the roots = a₀/a_nⁿ

Where n is the degree of the polynomial

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Polynomial P(x) has one root a+bi, with a and b real numbers, what is the other root?

The conjugate a-bi is also a root of P(x)

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What is |a+bi|?

(a² + b²)^1/2

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What does the discriminant D tell us for:

D > 0
D = 0
D <0

Roots are real and unequal
Roots are real and equal
Roots are imaginary (no real roots)

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For two Linear functions:

y = m₁x + b₁ and y = m₂x + b₂

If m₁ = m₂and b₁ not= b₂

The two lines are parallel

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For two Linear functions:

y = m1x + b1 and y = m2x + b2

If m₁ = m₂ and b₁ = b₂

Then the two lines coincide

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For two Linear functions:

y = m₁x + b₁ and y = m₂x + b₂

If m₁.m₂ = -1

Then these two lines are perpendicular

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For two Linear functions:

y = m1x + b1 and y = m2x + b2

If m₁ not= m₂

Then these two lines are intersecting

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Distance between point (x₁, y₁) and a line ax + by + c = 0

D = |ax₁ + by₁ + c|/(a² + b²)^1/2

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Distance between (x_1, y₁) and (x₂, y₂)

D = {(x₂-x₁)² + (y₂-y₁)²}^1/2

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Distance from the origin to a point (a, b, c)

[a² + b2 +c2]1/2

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Standard equation of a circle with center at (h, k) and radius r

(x-h)² + (y-k)² = r²

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Standard equation of an ellipse with center (h, k) and where a > b

(x - h)²/a² + (y - k)²/b² = 1 Major axis is horizontal

(x - h)²/b² + (y - k)²/a² = 1 Major axis is vertical

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Length of Major and Minor Axes of ellipse?

Major axis = 2a

Minor Axis = 2b

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For an ellipse if c is the length from the center to the focus what is its value?

c² = a² - b²

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Standard form of parabola with vertex at (0, 0)

Vertical axis: x² = 4py

Horizontal axis: y² = 4px

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Standard form of a hyperbola, center at (0, 0)

Transverse axis horizontal: x²/a² - y²/b² = 1

Transverse axis vertical y²/a² - x²/b² = 1

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Hyperbola Focus for (_+_c, 0)

c² = a² + b²

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Asymptotes, horizontal and vertical axis for a hyperbola

Horizontal: y = +(b/a)x

Vertical: y = +(a/b)x

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What are Domain and Range

Domain is the set of X (input)

Range is the set of Y (output)

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What are odd and even functions

Even function: f(x) = f(-x)

Odd function: f(x) = -f(-x)

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If p is the period of f(x) then:

f(x + p) = f(x)

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If p is the period of f(x) then what is the period of y = cf(x) y = f(cx)

Period is p Period is p/c

If g is the inverse of f, what are the properties?

f(g(x)) = x and g(f(x)) = x f^-1(x) is the reflection of the graph of f in the line y = x If point (a, b) lies on the graph of f, then the point (b, a) lies on the graph of f^-1

When does an inverse function exist?

When f is increasing on its entire domain When f is decreasing on its entire domain Use horizontal line test

If f is continuous on a closed interval [a ,b] and k is any number between f(a) and f(b) then

There is at least one number c in [a ,b] such that f(c) = k

If a polynomial has integer coefficients the possible rational zeros of f are:

(factors or constant term)/(factors of leading coefficient)

For a polynomial with real coefficients and a₀ not= 0

1. The number of **positive zeros** of f is either equal to the number of variations in sign of f or less than the number by an even integer 2. The number of **negative zeros** of f is either equal to the number of variations in sign of f or less than the number by an even integer

x → 0 lim (1 + x)^1/x and x → infinity lim (1 + 1/x)^x

What is the nth term t_n if the first is t₁ and the common difference is d

t_n = t₁ + d(n - 1)

**The sum** of a finite arithmetic sequence with n terms is

S_n = n(t₁ + t_n)/2

What is the nth term if the first term is t1 and the ommon ratio is r?

t_n = t₁r^n-1

Sum of the sequence?

S_n = t₁(1-rⁿ)/(1-r)

If |r| \< 1, the sum S of the infinite series is

S = t₁/(1-r)

A **permutation** of a set of values is **an arangement**, where **order is important**. The number of permutations of r elements from n elements is

_nP_r = n!/(n-r)!

A selection where **order is not important** is called a **combination** The number of combinations of n things taken r at a time

_nC_r = n!/(n-r)!r! or _nC_r = _nP_r/r!

The number of terms of (x + y)ⁿ

n + 1

The rth term of the expansion is

_nC_r-1(x)^n-r+1 (y)^r-1

Multiplication law: x^a_*x^b

X^a+b

Power Law: (x^a)^b

x^ab

Division Law: x^a ÷ x^b

x^a-b

Power of a Product Law: (xy)^a

x^a _* y^a

What is the compounding equation? continuous compounding equation?

A = A₀(1+r/n)^nt continuous compounding: A = Pe^rt

n \> 0, a \> 0 and a not 1 log_an = e then

a^e = n

log_a1=

log_aa =

log_a a^x =

a^log_ax =

log_a (xy)

log_ax + log_ay

log_a (x/y)

log_ax - log_ay

log_axⁿ =

nlog_ax

Sine-Cosine Identity

sin²x+cos²x=1

What are the domain, range and period of sinx?

Domain: All real numbers Range: -1\<=sinx \<=1 Period: 2pi, 360 deg

What are the domain, range and period of cosx?

Domain: All real numbers ## Footnote Range: -1\<=cosx \<=1 Period: 2pi, 360 deg

What are the domain, range and period of tanx?

Domain: All real numbers except x = 180n + 90 ## Footnote Range: all real numbers Period: pi, 180 deg

What are the domain, range and period of csc(x)?

Domain: All real numbers except 180n Range: |csc(x)|\>=1 Period: 2pi, 360

What are the domain, range and period of sec(x)?

Domain: All real numbes except 90 + 180n Range: |sec(x)| \>= 1 Perikod: 2pi, 360

What are the domain, range and period of cot(x)?

Domain: All real numbers except 180n Range: All real numbers Period: pi, 180

What is a cofunction?

Any trignometric function of an acute angle is equal to the cofunction of its complement

If A + B = 90 then sinA =

cosB

If A + B = 90 then what id the cofunction of secA

cscB

If A + B = 90 then what is the cofunction of tanA

cotB

For either y = asin(bx - c) + d or y = acos(bx-c) + d what are the amplitude, Period and Middle line?

Amplitude = |a| Period p = 2pi/b Middle line: y = d

For y = tan(bx) what is the Period?

Period: pi/2

What are the domain and range of arcsin(x)?

Domain: -1 =\0 =\0

What are the domain and range of arccos(x)

Domain: -1 =\0 =\0

What are the domain and range of arctan(x)?

Domain: all real numbers Range: -90⁰\0

sin(A + B) =

sinAcosB + cosAsinB

cos(A + B) =

cosAcosB - sinAsinB

tan(A + B) =

(tanA + tanB)/(1 - tanAtanB)

sin (A - B) =

sinAcosB - cosAsinB

cos (A - B) =

cosAcosB + sinAsinB

tan (A - B) =

(tanA - tanB)/(1 + tanAtanB)

sin2A =

2sinAcosA

cos2A =

cos²A - sin²A 1 - 2sin²A 2cos²A - 1

tan2A =

2tanA/(1 - tan²A)

sin(A/2) =

_+_ [(1-cosA)/2]^1/2

cos(A/2) =

_+_ [(1 + cosA)/2]^1/2

tan(A/2) =

_+_ [(1 - cosA)/(1 + cosA)]^1/2

If ABC is a triangle with sides *a, b,* and *c* then

*a*/sinA = *b*/sinB = *c*/sinC

If ABC is a triangle with sides *a, b,* and *c* then the area =

(*bc*sinA)/2 (*ab*sinC)/2 (*ac*sinB)/2

If ABC is a triangle with sides *a, b,* and *c* then what is the Law of Cosines

cosA =( *b*² + *c*² - *a*²)/2*bc* cosB = (*a*² + *c*² - *b*²)/2*ac* cosC = (*a*² + *b*² -*c*²)/2*ab*

For a vector V→ what is a unit vector u→?

u→ = (V→)/|V→| Divide a vector by its length

SAT 2 Subject test Flashcards

(85 cards)