SAT Math II

T1: ex. x^2+2x+6= a(x-1)^2+b(x-1)+c, what are the values of a and b?

a=1, b=4, c=1

T6: ex. If a+b+(a-b)i=6-4i, What are the values of a and b?

a=1, b=5

T1: ex. if ax-b= 3x+2, what are the values of a and b?

a=3, b=-2

T1: ex. a(x+1)= b(x-1)= x+9, what are the values of a and b?

a=5, b=-4

T7: ex. If y= 3x^2-2x+k is positive for all x, then what is the smallest integral value of k?

k must be >1/3, so k=1

T8: ex. If the two lines 2x-3y+2=0 and 3x-ky-1=0 are perpendicular, then k=

k=-2

T7: ex. If the roots of x^2+(k-1)x+4=0 are real and equal, what is the value of k?

k=-5 or 3

T3: ex. If a polynomial P(x)= x^2+kx-8 has a factor of (x-2), then what is the value of constant k?

k=2

T5: ex. If one of the roots of mx^2+(4m-1)x+k=0 is -1-2i, where m and k are real numbers, then what is the value of k?

k=5/2

T9: What is the distance between two parallel lines 3x+y=12 and mx+2y=4

sqrt(10)

T6: ex. What is the value of |3-4i|?

sqrt(9+16)=5

T5: ex. If one of the roots of a quadratic equation f(x)=0 is -3+2i, what is the quadratic equation?

x^2+6x+13

T4: ex. If one of the roots of a quadratic equation 2-i. what is the equation?

x^2-4x+5

T8: ex. What is the equation of the line which is equidistant from A(4,0) and B(0,2)?

y=2x-3

T2: Remainder Theorem

When a polynomial P(x) is divided by (x-a), the remainder R is equal to P(a). Polynomial P(x) can be expressed as follows: P(x)= (x-a)Q(x)+R

T3: ex. If a polynomail f(x)= x^3+ax^2+bx+1 has a factor of (x^2-1), what are the values of a and b?

a=-1, b=-1

T8: ex. If the two lines 2x+ay=1 and ax+(a+4)y=2 are parallel, then a=

a=-2 (4 is extraneous)

T6: ex. What are the additive inverse and multiplicative inverse of the complex number 3-i?

(3+i)/10

T7: ex. If a quadratic equation 2x^2-kx+3=0 have imaginary roots, what is the value of k?

-2sqrt6<k<2sqrt6

T4: ex. What is the sum of all zeros of a polynomial function P(x)= 2x^7+3x^3-5x^2+4?

0

T2: ex. g(x)= x^3+2x^2+2x+3 is divided by (x-1)(x-2), then waht is the remainder

15x-7

T4: ex. What is the product of all zeros of g(x)=3x^7-5x^3+3x^2+x-2

2/3

T10: ex. What is the distance from a point (1,2,-3) to a plane 3x-4y+12z=-2

3

T9: ex. What is the distance from a line 12x-5y=0 to a point (7,9)?

3 [formula:D=|ax1+by1+c|/(sqrt(a^2+b^2)]

T4: ex. If the roots of a quadratic equation 2x^2+5x-4=0 are alpha and beta, what is the value of 1/alpha + 1/beta

5/4

T9: ex. What is the distance from the origin to the line 3x+4y=8

8/5

T1: Identical Equations

An identical equation is an equation which is true for all values of a variable 10x+5x= 15x is an identical equation because it is true for all real x 10x+5= 15 is an algebraic equation because it is true for x=1 only

T6: Complex Number

Complex number= real+imaginary numbers Pure Imaginary number= only imaginary numbers 1. a+bi is complex number, where a and b are real numbers (standard form) 2. a-bi is the conjugate of a+bi 3. If a+bi=c+di, then a=c and b=d (equality of complex numbers) 4. |a+bi|=sqrt(a^2+b^2) (distance from the origin)

T9: Distance between a point and a line

D is the distance between a point (x1, y1) and a line ax+by+c=0: D=|ax1+by1+c|/(sqrt(a^2+b^2) Distance between two points D=sqrt((x2-x1)^2+(y2-y1)^2)

T7: Discriminant

Discriminant determines the nature of the roots of a quadratic equation ax^2+bx+c=0 Discrimant D=b^2-4ac 1. If D>0, then the roots are real and unequal (graph intersects at two points) 2. If D=0, then the roots are real and equal (graph touches x-axis at one point) 3. If D<0, then the roots are imaginary (No real roots) (graph floats above x-axis)

T10: Distance in Three Dimensions

Distance from point A(x2,y2,z2) to point B(x1,y1,z1) in space D=sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2) Distance from a point (x1,y1,z2) to a plane ax+by+cz+d=0, D=|ax1+by1+cz1+d|/sqrt(a^2+b^2+c^2) The distance from the origin to a point (a,b,c) is D=sqrt(a^2+b^2+c^2)

T4: Sum and Product of the Roots

For a polynomial P(x)=anx^(n)+an-1x^(n-1)+an-2x^(n-2)+...+a1x+a0=0 Sum of the roots=-an-1/an Product of the roots=a0/an*(-1)^n

T8: Linear Function

For two linear functions: y=m1x+b1 and y=m2x+b2 1. If m1=m2 and b1 does not equal b2, then these two lines are parallel (inconsistent) 2. If m1=m2 and b1=b2, then these two lines coincide (dependent) 3. If m1*m2=-1, then these two lines are perpendicular 4. If m1 is not equal to m2, then these two lines are intersecting (consistent)

T5: Conjugate Roots

If a polynomial function P(x) has one variable with real coefficients, and a+bi is a root with a and b real numbers, then its conjugate a-bi is also a root of P(x)

T3: Factor Theorem

If f(a)=0, then f(x) has a factor of (x-a) Polynomial f(x) can be expressed with a factor of (x-a) as follows: f(x)= (x-a)Q(x) Therefore, f(a)=0 means that the remainder is 0

T2: ex. f(x)= 2x^2-3x+5 is divided (x-1), what is the remainder?

R=4