Manhattan Algebra 2 Tutor

Manhattan Algebra 2 Tutor

Math Tutors

Cornerstone Protection IconCornerstone Protection

836 Euclid Avenue Suite 16, Lexington

5.0 37 reviews

  • Avatar Larry Barton ★★★★★ a month ago
    Amazing assistance provided by Shannon today on my security system. I’m so grateful for your prompt, efficient, professional, and courteous service. Put me down as a customer for life!
  • Avatar Megan Haller ★★★★★ 2 months ago
    One of the most painless experiences I've had. They came on time and was over the moon professional with still keeping a level of personability to it. The gentleman also went out of his way to say that my husband and Mines business … More was appreciated. Which is actually rare to be appreciated by such a great company. So thank you and we look forward to continuing services with this company.
  • Avatar Megan Magsam ★★★★★ 3 months ago
    I am so happy we decided to go with cornerstone. They were quick and efficient. Shannon and the installer we very friendly. We got a lot of equipment and it was still within our budget. 10/10 would definitely recommend! Thank you
  • Avatar Collins Youngblood ★★★★★ 2 weeks ago
    Love the customer service and Shanon and his team go above and beyond for my restaurant. Couldn't of asked for a better security system and level of service that his team provides.
  • Avatar kanu patel ★★★★★ 2 months ago
    I have been using Cornerstone Protection service for over 5 years and I'm really happy with the service and state-of-the-art equipments and it's futures are so much in advance and friendly, I highly recommend Cornerstone Protection … More Service any and all business that need a security protection days and nights specially after business hours(Cornerstone Protection is perfect solution for your business). Owner of Cornerstone Protection is proactive with their customers and their equipments. Good experienced with Cornerstone Protection!!!
    Kanu Patel
  • Avatar Liz Jones ★★★★ 8 months ago
    We had a very positive experience with Alissa as a tutor for our pod of 5th grade boys from January - June 2020. She cared a lot about the boys and worked hard to create meaningful activities for them. The pod experience was invaluable … More for all of our families during a difficult year and we were so grateful to have found this partnership! We also want to appreciate Craig for helping to match us with a tutor on very short notice after we had cycled through a few folks who were not able to see the commitment through. Thanks for sticking with us!
  • Avatar S CL ★★★★★ 11 months ago
    Dusty was the exact person we needed to help our child finish out the school year. So grateful to him for the way he steadily encouraged her and helped her stay on track and organized. He is flexible, responsive to our needs/schedule as … More a family, and a delight to be with. We are also thankful that Craig at Themba Tutors really "gets it" and partnered us with such a wonderful tutor who was a great fit for our girl.
  • Avatar Nicole May ★★★★★ 11 months ago
    My child worked with Gillian D virtually for math enrichment. She is an excellent teacher. She took time to understand my child's specific learning style and provided tools that would be engaging. She was kind and empathetic- really … More made it her goal to connect with my child and build trust and rapport. My child started the school year with confidence and without the typically "brain-drain" that occurs over the summer, thanks to Gillian.
  • Avatar Gillian Davidoff ★★★★★ 11 months ago
    I am going into my second year as a Themba tutor. I have met so many great individuals while working for this company. Craig and the rest of the Themba team do such an excellent job picking and choosing Themba tutors that suit each client's … More individual needs! I look forward to meeting new tutors and clients in the 2021-2022 school year!
  • Avatar carolina luttmann ★★★★★ 11 months ago
    Our son started in a new high school, 100% remote, and was really struggling with Latin and Physics. I found Themba Tudors online and had an initial call with Craig, who was very knowledgeable, responsive, and found 2 amazing tutors for … More us. Both the Latin and the physics tutors were fantastic, connected with my son really well and helped him go from 'failing' grade to 'honors'! I will for sure continue using (and recommending!) Themba Tutors. Thank you, C, K, and W!
  • Avatar Eric Fingerman ★★★★★ 10 months ago
    Brian is a fantastic executive functioning coach. Super helpful, flexible with hours, and tailored the experience to our needs.
  • Avatar Susan Wong ★★★★★ a year ago
    Giulia Mercuri is a great Italian tutor! She was very responsive and accessible. Giulia put a ton of effort into preparing for each session and really helped our son with test preparation and projects. We are grateful that she was able … More to help, especially since our son did not receive any live teaching during COVID. We are happy that she was referred to us!
  • Avatar Christine Gennetti ★★★★★ a year ago
    As an adult student diagnosed with a learning disability during grad school, I knew I needed help dealing with my Master's program's demanding workload. I was seriously considering dropping out when I was referred to Themba Tutors. … More I was paired up with a wonderful tutor, Julie, who was a total lifesaver. I learned many techniques that I can utilize in the future as I embark on the next phase of my career.
    I can wholeheartedly say after working with Themba for two years that I would not have finished my program without this organization. And I am so thankful I selected them.
  • Avatar Fabiola FITZPATRICK ★★★★★ 2 years ago
    We were introduced to a wonderful tutor: Mcedwyn. He and our son, a freshman in college, worked well together. As a result, our son’s grades significantly improved. The tutor taught our son subject content and various methods to effectively … More study for quizzes and exams. Our son was also taught time management and organization. My son’s experience with the tutor made the first year of college a less intimidating experience.
  • Avatar Jude Eugene ★★★★★ a year ago
    We have been with our tutor, Cailin Schiller, for two years, from 5th - 7th grade. She started as an executive functioning tutor with my daughter. From the start, she demonstrated a depth of expertise and a selective use of techniques skillfully … More adapted to my daughter’s ADHD and Math disability. Cailin made Math accessible and understandable to her in a way that even her teachers had not been able to do. She differentiated reading, writing and math tasks and assignments to tailor them to my daughter’s unique learning differences. Not only did it have an effect on her grades and her work efficiency, but she finally could do her work with less anxiety. Even my daughter’s teachers used Cailin’s guidance to modify their work with her. But most importantly, she became my daughter’s favorite resource. I can’t say how invaluable Cailin has been to my daughter’s progress and well-being. I’d recommend her a million times over. She is a gem.
  • Avatar Janice Eidus ★★★★★ a year ago
    Victoria is a superb tutor. She helped my high school age daughter with her writing. Victoria's skills as a tutor, as well as her sense of humor and her engaging personality went a long way with my daughter.
  • A Google User ★★★★★ 3 years ago
    I'm so glad we were referred to Themba Tutors. After a great initial conversation to learn our needs, we were connected with an excellent tutor, Felicia, who is helping our high school daughter learn better organizational and executive … More skills. Highly recommended.
  • Avatar Fara Jones ★★★★★ 3 years ago
    Themba Tutors provided a leaning specialist that was a perfect fit for my daughter. I told them what we were looking for and they delivered. Our tutor is responsive and flexible and makes it easy for my daughter to reach out to her own … More her own, which gives her an added sense of responsibility over her learning. As a therapist myself, I now regularly recommend Themba to my patients. A much needed and well provided service to the community!!
  • Avatar Frances Simpson ★★★★★ 3 years ago
    The tutors at Themba are fantastic! Caitlyn McNeill helped our son academically, but also with getting organized in every subject, which he so desperately needed. She offered great advice and useful strategies that helped our son become … More a more successful student. We are using Themba tutors again this year for our daughter. We're so happy to have been referred to them!
  • Avatar Mary Jean Babic ★★★★★ 3 years ago
    I'm so glad we were referred to Themba Tutors. After a great initial conversation to learn our needs, we were connected with an excellent tutor, Felicia, who is helping our high school daughter learn better organizational and executive … More skills. Highly recommended.
  • Avatar Nina Herman ★★★★★ 3 years ago
    We had a wonderful experience with our tutor Micaela. She helped our son not only academically but at every level. We will be forever grateful to have her in our lives.
  • Avatar Frances ★★★★★ 3 years ago
    The tutors at Themba are fantastic! Caitlyn McNeill helped our son academically, but also with getting organized in every subject, which he so desperately needed. She offered great advice and useful strategies that helped our son become … More a more successful student. We are using Themba tutors again this year for our daughter. We're so happy to have been referred to them!
  • Avatar Michael Birnbaum ★★★★★ 3 years ago
    Paul Smith at Themba has been an exceptional tutor for my son, both in terms of teaching the subject matter and teaching how best to approach the process of learning math and executing assignments. We feel very fortunate to be working with … More Paul.
  • Avatar Monica Alzate ★★★★★ 3 years ago
    We were very lucky to have found Edwouine Swift, my son's therapist through Themba Tutors. Thomas is 12 years old, and it is extremely difficult to get him to write. Edwouine just works her magic with him, he adores her and is so … More eager to follow her guidance. I have no doubt she will be a positive driving force in his academic future.

Top Math Tutors in Brooklyn

MANHATTAN ALGEBRA 2 TUTOR

Manhattan Algebra 2 Tutor

WE travel to YOUR NYC home:

Brookyn Manhattan Queens Staten Island The Bronx

WE also travel to:

Westchester New Jersey (NJ) Connecticut Long Island

Module 1: Polynomial, Rational, and Radical Relationships

A: Introduction to Polynomials

A polynomial is an expression composed of variables, constants, and exponents that are combined using mathematical operations such as addition, subtraction, multiplication, and division (No division operation by a variable).

A polynomial looks like this:

Manhattan Algebra 2 Tutor

an example of a polynomial with 3 terms

Polynomials with one variable make nice smooth curves:

manhattan algebra 2 tutorA polynomial can have:

constants (like 3, −20, or ½)
variables (like x and y)
exponents (like the 2 in y2), but only 0, 1, 2, 3, … etc are allowed
that can be combined using addition, subtraction, multiplication, and division.
A polynomial can have constants, variables, and exponents, but never division by a variable. Also, they can have one or more terms but not an infinite number of terms.

Polynomial or Not?

manhattan algebra 2 tutor

These are polynomials:
3x
x − 2
−6y2 − (7/9)x
3xyz + 3xy2z − 0.1xz − 200y + 0.5
512v5 + 99w5
5

(Yes, “5” is a polynomial, one term is allowed, and it can be just a constant!)

These are not polynomials
3xy-2 is not, because the exponent is “-2” (exponents can only be 0,1,2,…)
2/(x+2) is not, because dividing by a variable is not allowed
1/x is not either
√x is not, because the exponent is “½” (see fractional exponents)

But these are allowed:
x/2 is allowed, because you can divide by a constant
also 3x/8 for the same reason
√2 is allowed, because it is a constant (= 1.4142…etc)

Monomial, Binomial, Trinomial

There are special names for polynomials with 1, 2 or 3 terms:

manhattan algebra 2 tutor

How do you remember the names? Think cycles!

manhattan algebra 2 tutor

There is also quadrinomial (4 terms) and quintinomial (5 terms), but those names are not often used.

Variables

Polynomials can have no variable at all.
Example: 21 is a polynomial. It has just one term, which is a constant.
They may have one variable
Example: x4 − 2×2 + x has three terms, but only one variable (x)
Or two or more variables
Example: xy4 − 5x2z has two terms, and three variables (x, y and z)

What is Special About Polynomials?

Because of the strict definition, polynomials are easy to work with.
For example, we know that:
If you add polynomials you get a polynomial
If you multiply polynomials you get a polynomial

So you can do lots of additions and multiplications, and still have a polynomial as the result. Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines.

Example: x4−2×2+x

manhattan algebra 2 tutor

See how nice and smooth the curve is?

You can also divide polynomials (but the result may not be a polynomial).

Degree

The degree of a polynomial with only one variable is the largest exponent of that variable.

Example:
4×3 − x + 2

The Degree is 3 (the largest exponent of x)

Standard Form

The Standard Form for writing a polynomial is to put the terms with the highest degree first.
Example: Put this in Standard Form: 3×2 − 7 + 4×3 + x6

The highest degree is 6, so that goes first, then 3, 2, and then the constant last:
x6 + 4×3 + 3×2 − 7

Common Symbols Used in Algebra

Symbol

+

×

·

/

 

 

( )

[ ]

{ }

=

< ≤

> ≥

!

Meaning

add

subtract

multiply

multiply (as “×” can look like “x”)

divide

square root (“radical”)

cube root

nth root

grouping symbols (round brackets)

grouping symbols (square brackets)

set symbols (curly brackets)

equals

approximately equal to

not equal to

less than, less than or equal to

greater than, greater than or equal to

implies (if … then)

“if and only if” or iff or “is equivalent to”

Therefore

Factorial

Example

3+7 = 10

5−2 = 3

4×3 = 12

4·3 = 12

20/5 = 4

√4 = 2

∛8 = 2

 

2(a−3)

2[ a−3(b+c) ]

{1,2,3}

1+1 = 2

π ≈ 3.14

π ≠ 2

2 < 3

5 > 1

a and b are odd ⇒ a+b is even

x=y+1 ⇔ y=x−1

a=b ∴ b=a

4! = 4×3×2×1 = 24

Topic B: Adding and Subtracting Polynomials

A polynomial looks like this:

manhattan algebra 2 tutor

example of a polynomial
this one has 3 terms

To add polynomials we simply add any like terms together. Like Terms are terms whose variables (and their exponents, such as the 2 in x2) are the same.

In other words, terms that are “like” each other.

Note: the coefficients (the numbers you multiply by, such as “5” in 5x) can be different.

Example:
7x                   x                   -2x                   πx

are all like terms because the variables are all x

Example:
(1/3)xy2                   -2xy2                   6xy2                   xy2/2

are all like terms because the variables are all xy2

Example: These are NOT like terms because the variables and/or their exponents are different:
2x                   2x2                   2y                   2xy

Adding Polynomials

 

Step 1: Arrange each polynomial with the term with the highest degree first then in decreasing order of degree.
Step 2: Group the like terms. Like terms are terms whose variables and exponents are the same.
Step 3: Simplify by combining like terms.

Example: Add 2×2 + 6x + 5 and 3×2 – 2x – 1

Start with:
2×2 + 6x + 5 + 3×2 − 2x − 1

Place like terms together:
2×2+3×2 + 6x−2x + 5−1

Which is:
(2+3)x2 + (6−2)x + (5−1)

Add the like terms:
5×2 + 4x + 4

Other Examples:

manhattan algebra 2 tutor

(adding in columns)
Adding Several Polynomials – Using columns helps us to match the correct terms together in a complicated sum.

Example: Add (2×2 + 6y + 3xy) , (3×2 – 5xy – x) and (6xy + 5)

Line them up in columns and add:
2×2 + 6y + 3xy
3×2 – 5xy – x
6xy + 5
5×2 + 6y + 4xy – x + 5

Subtracting Polynomials

To subtract Polynomials, first reverse the sign of each term we are subtracting (in other words turn “+” into “-” and “-” into “+”), then add as usual.

manhattan algebra 2 tutor

Note: After subtracting 2xy from 2xy we ended up with 0, so there is no need to mention the “xy” term anymore.

Topic C: Multiplying Polynomials

To be able to multiply polynomials,
First, multiply each term in one polynomial by each term in the other polynomial using the distributive law.
Add the powers of the same variables using the exponent rule.
Then, simplify the resulting polynomial by adding or subtracting the like terms.

example of a polynomial
this one has 3 terms

manhattan algebra 2 tutor

1 term × 1 term (monomial times monomial)
To multiply one term by another term, first multiply the constants, then multiply each variable together and combine the result, like this (press play):

Manhattan Algebra 2 Tutor

(Note: I used “·” to mean multiply. In Algebra we don’t like to use “×” because it looks too much like the letter “x”)

1 term × 2 terms (monomial times binomial)
Multiply the single term by each of the two terms, like this:

Manhattan Algebra 2 Tutor

2 term × 1 terms (binomial times monomial)
Multiply each of the two terms by the single term, like this:

Manhattan Algebra 2 Tutor

(I did that one a bit faster by multiplying in my head before writing it down)

2 terms × 2 terms (binomial times binomial)
We can multiply them in any order so long as each of the first two terms gets multiplied by each of the second two terms. But there is a handy way to help us remember to multiply each term called “FOIL”.

It stands for “Firsts, Outers, Inners, Lasts”:

Manhattan Algebra 2 Tutor

Firsts: ac
Outers: ad
Inners: bc
Lasts: bd

So you multiply the “Firsts” (the first terms of both polynomials), then the “Outers”, etc. Let us try this on a more complicated example:

Manhattan Algebra 2 Tutor

2 terms × 3 terms (binomial times trinomial)
“FOIL” won’t work here, because there are more terms now. But just remember: Multiply each term in the first polynomial by each term in the second polynomial

Manhattan Algebra 2 Tutor

Like Terms

And always remember to add Like Terms:

Example: (x + 2y)(3x − 4y + 5)
(x + 2y)(3x − 4y + 5)
= 3×2 − 4xy + 5x + 6xy − 8y2 + 10y
= 3×2 + 2xy + 5x − 8y2 + 10y

Note: −4xy and 6xy are added because they are Like Terms.

Also note: 6yx means the same thing as 6xy

Long Method

Choose one polynomial (the longest is a good choice) and then:

  • multiply it by the first term of the other polynomial, writing the result down
  • then multiply it by the second term of the other polynomial, writing the result under the matching terms from the first multiplication
  • then multiply it by the third term of the other polynomial (if any), etc.
  • lastly, add up the columns.

By lining up the columns and being careful to put the terms under the correct columns, the job becomes “automatic”, and we can easily look back to see if we got it right, too.

But what happens if a polynomial is missing, say, an x term or an x2 term? Just leave that column blank!

Here is a more complicated example with blank gaps:

Manhattan Algebra 2 Tutor

More than One Variable
So far, we have been multiplying polynomials with only one variable (x), but how do we handle polynomials with two or more variables (such as x and y)? What are the column headings?
Just ignore the columns in the question, and write down the answers as they come, always checking to see if we could put an answer under a matching answer:

Topic D: Dividing Polynomials

Sometimes it is easy to divide a polynomial by splitting it at the “+” and “−” signs, like this (press play):

Manhattan Algebra 2 Tutor

When the polynomial was split into two parts we still had to keep the “/3” under each one.

Then the highlighted parts were “reduced” (6/3 = 2 and 3/3 = 1) to leave the answer of 2x-1

Here is another, slightly more complicated, example:

Manhattan Algebra 2 Tutor

What happened?

  • The 1st Term had x2 above and x below, which together becomes just x
  • The 2nd Term had x above and below, so they canceled each other out
  • We couldn’t simplify “1/3x” any further.

That is as far as we can get. But the answer is still “simpler”

Note: the result is a valid answer but is not a polynomial, because the last term (1/3x) has division by a variable (x).

Now, sometimes it helps to rearrange the top polynomial before dividing, as in this example:

Long Division

Dividing the Numerator by the Denominator
We can give each polynomial a name:

Manhattan Algebra 2 Tutor

  • the top polynomial is the numerator
  • the bottom polynomial is the denominator

If you have trouble remembering, think denominator is down-nominator.

Write it down neatly:

Manhattan Algebra 2 Tutor

  • the denominator goes first,
  • then a “)”,
  • then the numerator with a line above

Both polynomials should have the “higher order” terms first (those with the largest exponents, like the “2” in x2).

Then:

  • Divide the first term of the numerator by the first term of the denominator and put that in the answer.
  • Multiply the denominator by that answer, put that below the numerator
  • Subtract to create a new polynomial
  • Repeat, using the new polynomial It is easier to show with an example!

Example:

Write it down neatly like below, then solve it step-by-step:

Manhattan Algebra 2 Tutor

Check the answer:
Multiply the answer by the bottom polynomial, and we should get the top polynomial:

Manhattan Algebra 2 Tutor

Remainders

The previous example worked perfectly, but that is not always so! Try this one:

After dividing we were left with “2”, this is the “remainder”.

The remainder is what is left over after dividing.

But we still have an answer: put the remainder divided by the bottom polynomial as part of the answer, like this:

“Missing” Terms

There can be “missing terms” (example: there may be an x3, but no x2). In that case either leave gaps or include the missing terms with a coefficient of zero.

Example:

Manhattan Algebra 2 Tutor

Write it down with “0” coefficients for the missing terms, then solve it normally:

Manhattan Algebra 2 Tutor

More than One Variable

So far, we have been dividing polynomials with only one variable (x), but we can handle polynomials with two or more variables (such as x and y) using the same method.
Example:

Manhattan Algebra 2 Tutor

Solution:

Manhattan Algebra 2 Tutor

Students of all ages receive one-on-one guidance in all instructional materials, exam preparation, and executive function therapy. At Themba Tutors, we take the time to help students and clients appreciate their strengths and succeed in areas where they need to improve. We offer personalized tools for each of our clients to help them become more aware of their individual needs as they work toward short-term goals. These skills will assist our students/clients in achieving their long-term goals.

MEET OUR MANHATTAN ALGEBRA 2 TUTORS WHO TRAVELS TO YOUR HOME:

Manhattan Algebra 2 Tutor
OLGA

Master’s in Education, Bachelor’s in Secondary Math Education

remote ib math tutor
DAVID

Master’s in Mathematics Education (7-12)

upper east side math tutor
ALEX

Undergraduate Degree in Secondary Education with a Concentration in Mathematics, Graduate Degree in Special Education

EVAN

Master’s in Leadership in Math Education, Bachelor’s Degree in Mathematics

Manhattan Algebra 2 Tutor
MELISSA

Certified Math and Special Education Teacher, MS Math Education

**Guarantee: It’s critical that whoever receives our tutoring services connects with the specialist, the specialist has expertise in the area of need, and is available to travel to you. If you are not 100% satisfied with our services, we are 100% committed to finding you the right professional.

Chat with Themba Tutors Today! Our Manhattan Algebra 2 Tutors are ready to help you right now!

FREE CONSULTATION!!!

Call: (917) 382-8641, Text: (833) 565-2370 

Email: info@thembatutors.com

(we respond to email right away!).

References:

https://www.mathsisfun.com/algebra/index-2.html

The following two tabs change content below.
Avatar photo
Craig Selinger, CEO of Themba Tutors (serving Manhattan, Brooklyn, Queens, Staten Island, Bronx, Westchester, Long Island, New Jersey, and Connecticut), is a NY State licensed speech-language pathologist, executive functioning coach, and learning specialist with over 18 years of experience working professionally with over a thousand families. His expertise includes language-based learning issues, e.g. reading, writing, speaking, and listening, executive functioning, ADHD/ADD, and learning disabilities. Check out his interviews with top-notch professionals in the field on Spotify.
Share:
No Comments

Post A Comment