Month: April 2017

Week 11 – Guidelines for Factoring a Polynomial Expression

by nighinar2016

When factoring a polynomial, there are a couple of steps to take first when trying to factor it.

Always look for the GCF (greatest common factor) before anything else. Checking for a common factor and removing it is the first step. This GCF is for each monomial or the polynomial. When you take it out, place it in-front of the bracket.

For example: 3x^2 + 9x^2 – 30

The greatest common factor is 3, because they all share the GCF 3.

3(x^2 + 3x^2 – 10)

Now, we are either left with a binomial expression or a trinomial expression.

If it is a binomial expression, check to see if there is a difference of squares. Factor as such.

z^2 – 1

(z+1)(z-1)

If it is a trinomial expression, ax^2 + bx + c, a either = 1 or does not equal 1.

If a, or the leading coefficient, is a negative number, factor out the negative first.

If a = 1, we can use the method of inspection, finding two numbers that have the sum of the middle term and the product of the last term.

x^2 + 9x + 20

(x + 4)(x + 5)

4 + 5 = 9

4 x 5 = 20

If a does not = 1 and is higher or lower, we can use the method of guess and test or area diagram.

Then check if there is possibility for further factoring, checking to see if there is a difference of squares that can be factored.

We can use an acronym to remember all the steps when factoring polynomials.

CDPEU

Can Divers Pee Easily Underwater

Common factor Difference of squares Pattern Easy Ugly

C – look to see if there is a common factor

D – look to see if there is a difference of squares to factor

P – is it x^2, x, then a number or x^10, x^5, then a number

E – is the leading coefficient 1 and easy to factor by finding two terms that have the sum of the middle term and product of the last.

U – is it an ugly expression that takes more work to find the factoring

Then always check if there is possible further factoring.

Week 10 – Factoring Difference of Squares

by nighinar2016

A difference of squares if a factorization of (a – b)(a + b)

(a – b)(a + b) = a^2 + ab – ba – b^2

= a^2 – b^2

The middle two terms cancel out and you are left with a subtraction of the two terms squared, or a difference of squares.

So, we can apply this distributive property when factoring anything with the same terms but with different signs ( – and + )

Example

(10x – yz)(10x + yz)

Since this follows the rules, we can go straight to

100x^2 – y^2z^2

We can also work backwards with this factorization.

If we have something such as 144p^2q^2 – 4

We know that all those terms are squares.

The first step is to determine if there is a common factor that can be removed. In this case, that common factor is 4.

4(36p^2q^2 – 1)

Now, we are still left with a difference of squares that we can factor.

4(6pq + 1)(6pq – 1)

 

 

Week 9 – Three Important Polynomials Products

by nighinar2016

The distribution of the following polynomials

(a+b)^2 = (a+b)(a+b)

= a^2 + ab + ba + b^2

= a^2 +2ab + b^2

 

(a - b)^2 = (a – b)(a – b)

= a^2 – ab – ba + b^2

= a^2 – 2ab + b^2

 

(a – b)(a + b)

= a^2 + ab – ba – $latex b^24

= a^2 – b^2

When expanding and simplifying a polynomial, we can use the patterns or answers of the following polynomials to help us simplify quicker and easier.

So, when seeing the follow polynomials that fit the equation, you can go straight to the answer without having to go through all the distributive property.

This helps when simplifying longer polynomial expressions.

For example

Expand and simplify (x + 5)(x – 5) – (x + 2)(x+8)

We can go straight from (x + 5)(x – 5)  to  x^2 – 25 because we know that the terms are the same and we know the simplified answer without going through the long distribution.

Now we can continue to simplify the rest. You can’t use any of the three products from above here because (x + 2)(x+8) do not share all the same numbers.

= x^2 – 25 – (x^2 + 8x + 2x + 16)

= x^2 – 25 – (x^2 + 10x + 16)

Bring in the negative.

= x^2 – 25 – x^2 – 10x – 16

The x^2 cancels out because there is a negative and a positive one of the same thing. Continue simplifying and adding like terms.

= -10x – 41

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I found this lesson very useful. It was easy to remember once I got the hang of it and it helped me simplify polynomials quicker.

Week 8 – F.O.I.L

by nighinar2016

When multiplying two binomials, we can use a distributive property called F.O.I.L.

The distributive property for binomials: (a+b)(c+d) = a(c+d) +b(c+d) = ac + ad + bc + bd

Instead of going through the distributive property individually, we can use the simplified way, the F.O.I.L method.

The acronym:

F – first term in each bracket – ac

O – outside terms – ad

I – inside terms – bc

L – last term in each bracket – bd

Example:

–

A challenge with F.O.I.L is that it can become confusing if you forget what you have multiplied and what you haven’t. Writing down the arrows as you go can be very helpful and help you know where you are at and how much you have left.

Teens and Identity – Whirligig Introductory Assignment

by nighinar2016

Image result for stressed teen

When we were younger, everyone was friends. Life was easy, there was no judgement, and things were good. As we grew older, there came new stresses we never learned how to deal with. High school started, and your friends became distant. School work is more stressing and life at home might no longer be as easy as before. There is pressure on kids found everywhere. There’s pressure to fit in, look a certain way to be cool and to avoid judgement. Your personality and your interests must be refined and you must enjoy certain things. There is a competition in school work, you must be successful to graduate and get into a good school; you must be the smartest to be noticed by schools. You will always be reminded you are not as smart as “that kid” and that you never compare. Or maybe you are the smart kid, and you find that you are hated for it. There is pressure your family may put on you. They want you to be excelling in school to ensure you make something of yourself. You also have the sports and instruments you play or work you have on the weekends nagging you at side, reminding you that this is also something you must be good at and must practice or continue. Teens find themselves going through these different struggles in life; they become scared and alone, disoriented. We, as teens, have responsibilities and pressure that seem to build up and grow over time. This can become overwhelming and destructive if not dealt with properly. It’s natural to want to succeed in every area; friends, family, school, sports, work, social statuses; but if doesn’t work out it can lead to destruction.

We are all different, and our mental capacity to deal with what is thrown at us in our society is unique. Some people are lucky enough to have people that support them and help them through it all. Some people breakdown. They breakdown and they may grow stronger from it, or they may let it continue destroying them. Our society is so focused on statuses and how good you are at something; this pressure scares kids. They don’t feel they can be free to be themselves and they never get a break from the worry of stress. Its natural to crave an escape from this all. Even for a limited period. This outlet is usually drugs. Drugs give you an overdose of dopamine, a rush of happiness and carefree. The problem of this stress less time is that it can cause future stress. Possibly, you forget to do your homework, your family finds out and is disappointed, your coach finds out and kicks you off the team, your brain becomes dependent on drugs and you find you need it to feel happiness again. Self-medicating yourself with drugs, thinking they can help you with your issues in life, is toxic. You may feel great momentarily, put you are setting yourself up for further destruction.

I think teenagers need to learn to love themselves and surround themselves with those who love and support them for being themselves. By removing toxic relationships with those who judge them and have conditions to their acceptance, they allow themselves to find those who can help lift some of the pressures and stresses. Then, they can free up one of their worries, they no longer need to worry about appearance and can focus on sports, school, etc.

Week 7 – Calculating for Θ (Theta)

by nighinar2016

Theta is the symbol used for the unknown angle in a triangle. When working with right angle triangles – triangles with one 90 degree angle – you can use Soh Cah Toa to help you find the unknown angle.

Before learning how to find the missing angle right away, there’s a few steps to understand.

First, label your triangle. This will let you know which Trigonometry function to use when calculating Θ.

There’s the hypotenuse – the longest length of the right angle triangle, the opposite – the angle opposite to the Θ or the angle you are looking for, and the last length left is called the adjacent.

In order to find Θ, you need to know at least two lengths.  This can help you apply Soh Cah Toa.

Soh Cah Toa is an acronym to help you find with trig function you should use in order to find Θ. The first letter of each three is the trig function; S = sine, C = Cosine, T = tangent. The next two letters are the side lengths and their order; o = opposite, a = adjacent, h = hypotenuse. So; Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

When you know two lengths and you know their side names, you can find the function you need to solve for Θ. For example, if you know the length of the opposite and hypotenuse, you can use sine to find the angle.

When solving for an angle, you use the inverse of the function.

sin(Θ)= 20/29 

Θ = sin–1(20/29)

Θ = 44 degrees. (rounded to the nearest ten)

When finding a length when you know your angle, you can use one of the functions. For example if we know the opposite and the angle and are trying to find the hypotenuse, we can use sine again. x = the unknown length (hypotenuse)

sin(44 degrees) = 20/x

20 ÷ sin(44 degrees) = x

29 (rounded to the nearest ten) = x

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I found trig to be a very easy subject of math. As long as you were able to identify the lengths of the triangle, there was an acronym to follow after and formulas that made finding the number simple.