Unit 3B- Complex Numbers, Factoring, Parabola

Yujia Shen, Jonathan Kurian, Ryan Okushi

Complex Numbers

A complex number is a number that can be expressed in the form a+bi, where a and b are real numbers and i is the imaginary unit.

i¹=i → when exponent is divided by 4 and has remainder of 1, ix=ii²= -1 → when exponent is divided by 4 and has remainder of 2, ix=-1i³= -i → when exponent is divided by 4 and has remainder of 3, ix=-ii4=1 → when exponent is divided evenly by 4, ix=1

i =

+/-/x/÷ Complex Numbers

When adding and subtracting complex numbers, treat i as you would a variable.(-2+3i)+(3-2i)= -2+3+3i-2i=1+i(-2+3i)-(3-2i)= -2-3+3i+2i= -5+5i

“ Find all solutions to x4-16=0

x=2 or x=-2

x²=-4

or

Factoring

Factoring is a very important part of this unit. Factoring is breaking down a number or polynomial into parts that, when all multiplied together, will equal the original number or polynomial.

When factoring regular numbers, we break them down into prime numbers.

Ex. 12 will be broken down into 12= (2)(2)(3) &

112 will be broken down into (2)(2)(2)(2)(7)

Factoring (Cont.)

We do the same thing with polynomials: break them down into parts that can not be broken down anymore.

Ex. x2+12x+35=(x+7)(x+5)3x2+21x+36=3(x+4)(x+3)

The standard form of a trinomial is ax2+bx+c

How to Factor: The GCF

For factoring polynomials, finding the GCF is necessary in most methods. First, you need to check if all the terms have a GCF. If they do, take it out of the equation to make it easier.

3x2+21x+36=3(x2+7x+12)

3 could be taken out of all 3 terms, since it was the GCF of all of them

If it is 3x2y4+9x5y2, the GCF would be 3x2y2. You would take the GCF out of the equation, and make it 3x2y2(y2+3x3). That would be the factored form of the equation.

How to Factor: Busting up the b

2x2+7x+6

In the busting up the b, first factor out the GCF if there is one.

Make sure a is positive (multiply the equation by -1 if not)

Then multiply a and c. Find factors that when added up, equal b

2*6=12, 4+3=7 4 and 3 are the factors

Then put it into the (ax+ )(ax+ ) equation

(2x+3)(x+2)=2x2+7x+6

How to Factor: Grouping

The grouping method is typically used with 4 terms. Ex. 2x2+4x+3x+6

Order the terms in decreasing order by exponents, if they already aren’t.

Then group the first 2 and last 2 together. (2x2+4x)+(3x+6)

Then factor the GCF out of each. 2x(x+2)+3(x+2)

Add the factored part together and multiply by the inside of the parenthesis

(2x+3)(x+2)

Ex.3x³+9x2+x+3

(3x³+9x2)+(x+3)

3x2(x+3)+1(x+3)

(3x2+1)(x+3)

Special Cases

There are some special cases when factoring, such as squaring the binomial. Ex. (x+2)2 = x^2+4x+4. The equation for squaring is

(x+y)2=x2+2xy+y2 or

(x-y)2=x2-2xy+y2

The other special case is where there is no b, called difference of squares. Ex. x2+81. The factors would be (x+9)(x-9) The equation for these kinds are

(x+y)(x-y)=x2+y2

Difference/Sum of Cubes

There are more unique polynomials, namely sum of cubes and differences of cubes. A sum of cubes is in the form of a3+b3. To factor it, the equation is

The difference of cubes is in the form of a3-b3. To factor it, the equation is

Completing the Square

Completing the square is turning the original ax2+bx+c into a(x+d)2 + e = 0. From there, you can find the different values of x. In order to complete the square, you must add (b/2)2.The picture below shows why this is called “completing the square.”

Next, you must subtract (b/2)2 from c in order to balance the equation. The new number will be e. Now you should have the equation a(x+d)2 + e = 0.

A much, much faster way is to remember d=b/2a, and e= c-(b2/4a).

Parabolas

Standard Form: y=ax²+bx+c (up/down)

x=ay²+by+c (left/right)

vertex: ((-b/2a, f(-b/2a))

Vertex Form: a(x-h)²+k (up/down)a(y-k)²+h (left/right)

(h,k) is the vertex

Converting Standard Form and Vertex Form

WRITE IN VERTEX FORMy=x²+12x+32y-32=x²+12xy-32+36=(x+6)²y+4=(x+6)²y=(x+6)²-4

Vertex: (-6,-4)

“ Write the vertex form of y=x²+16x+71

Focus and Directrix

A parabola is the locus of all points that are equidistant from a given point “focus” and a given line “directrix” c= distance between the focus and the vertex c= distance between the vertex and directrix How to find focus and directrix:1. Find vertex using x=-b/2a if up/down parabola, and y=-b/2a if left/right

parabola. Find other coordinate by plugging into equation2. Find c3. The focus is inside the parabola while the directrix is a line outside of

it.. Add or subtract ‘c’ to the vertex. If it is an up/down parabola, the y coordinate is changed, the focus is written in (x,y), and directrix is written in the form of y=. If it is a left/right parabola, the x coordinate is changed, the focus is written in (x,y), and directrix is written in the form of x=.

PARABOLAS IN THE REAL WORLD

Bob likes to play with balls.

He throws a basketball that makes the shape of a parabola with the equation of y=-2x²+5x, a baseball that makes the shape y=-x²+4x, and a tennis ball that makes the shape y=-5x²+8x. Which one goes higher?

Bob is standing on a hill 80 feet high. He throws a bowling ball upward with an initial velocity of 64 feet per second. The height of the ball h(t) in terms of the time t since the ball was thrown is h(t) = -16t2 + 64t + 80. Find the max height and when the bowling ball reaches the max height

k