Algebra Homework Help: How to Factor Polynomials

How to Factor Polynomials

If there is a constant in the math universe, it is the letter X. In algebra, X is the source of many a mental math meltdown. “Finding ” is the instruction in every single problem. Learn here to factor polynomials.

But X can be fun! We can play with it, move it around, multiply it, divide it, and cancel it out. X really lets us see the relationship between numbers and how math theorems work.

Totally Awesome, I know!

But I know the only reason you’re here is to learn about polynomials and what to do with them.

Factoring polynomials is a crucial step to using the quadratic formula. Factoring polynomials is easier, though, and faster, too.

Let’s look at some X factors.

3x, x, -x2, 5x2, 5x, 7x, 20, -4

The first thing we do is put them in descending order of x.

-x2, 5x2, 3x, x, 5x, 7x, 20, -4

We can simplify them by adding like-factors together: all of the x2, then all the x, then the numbers without x

4x2 + 16x + 16

The next step is to look for common factors in all three groups. For example, these groups all have even numbered coefficients, so we know that 2 is a factor. Even more significant, we can see that all these integers are divisible by 4!

Take out the four from all of them.

4(x2 + 4x + 4)

We’re almost there!

Let’s look at (x2 + 4x + 4) . Can we break this down anymore?

The answer is yes! With the FOIL system!

FOIL stands for “first, outside, inside, last”. This is a cute way of saying that we got this expression by multiplying two parts separate factors together. Each of these factors is called a Binomial. (This is because each factor has two parts: an unknown and a number).

Here we are:

(a + b) X (c + d) = (x2 + 4x + 4) .

Basically, we end up with: ac+ ad + bc + bd. When we add them all together, we get (x2 + 4x + 4).

In this case, bd = 4, the last part of our expression, while ac = x2, the first part of our expression. The other two parts, ad+ bc, need to equal 4x.

We know that ac = x2, so A and C must be x. So that’s easy.

The hard part is figuring out ad and bc. But to solve that, we can look to bd. What are factors that, when multiplied together, give us 4? 4 breaks down to 1, 2, and 4. That means B or D must be 1, 2, or 4.

*****Let’s try 1 and 4. Plug them in and see what happens
(x + 1)(x + 4)

Use FOIL to multiply them together: First, Outside, Inside, Last.

We get x2 + 4x + x + 4

Add them together: x2 + 5x + 4 ≠ x2 + 4x + 4

Yikes! That didn’t work.****

Then it must be 2. This actually makes sense, since 2 + 2 = 4 or 2x + 2x = 4x!

Let’s double check

(x +2)(x + 2) → x2 + 2x + 2x + 4 → x2 + 4x + 4 Bravo!

We can even make this simpler. Since (x + 2) is the same, we can just square the binomial (x + 2)2

So the final answer is 4(x + 2)2

Here’s a rough one: 3x3 + 15x2 – 36x

First thing is to take out the greatest common factor: 3x Check it and see!

3x(x2 + 5x – 12)

Now, we can FOIL the larger expression.

****Here’s a trick: look at the coefficients in the middle (5) and end (12). You know that two factors multiplied together will bring you the end number, but they must be added/subtracted to form the middle number 5.

What are the multiplication factors of 12? 1 and 12, 2 and 6, and 3 and 4. None of these sets of factors, when added or subtracted, give you 5.

If you don’t believe me, try finding the binomials through trial and error. You’ll be at it forever!

Rats! We’ve been “foiled!” We can’t break this down further!

The answer is
3x(x2 + 5x – 12)

One more: -2x2 + 17x – 36

First of all, are there any common factors? Unfortunately, no. One number is odd, so 2 is not a factor in all three. The last number has no x, so that can’t be a factor, either.

We’ll just proceed to the FOIL, then.

This is a bit hard, since we have THREE GROUPS of numbers to deal with: 2, 13, 36. But we can still do this! Just take your time.

What are the factors of 36 that can add up to 36?

36 is broken down to: 1 and 36, 2 and 18, 3 and 12, 4, and 9, and 6 and 6.

None of these really add up to 17, but we have to consider the 2. There are two factors that make up 2: 1 and 2. Can we multiply any of the factors that make up 36 by 1 and 2, and then add them together to get 17?

YES!

Look at factors 4 and 9. They match up with 1 and 2. Multiply and add the sums:

1 x 9 = 9
2 x 4 = 8
9 + 8 = 17!!! Hooray!

Since the 2 and the 4 are multiplied together, they must be in separate factors: (2x, 9)(x, 4)

Now to determine which ones are positive and negative.

This might sound harder than it is.

We know that two positives multiplied together form a positive. (5 x 4) = 20

Two negatives multiplied together form a positive. (-7 x -9) = 63

A positive and negative multiplied together form a negative. (-8 x 2) = -16

Go back to the original equation and our factors:
-2x2 + 17x – 36

(2x, 9)(x, 4)

We know the first and last coefficients are negative. That means one integer in each binomial must be negative and one must be positive so that, when multiplied, we get negative numbers. But we also know that the middle coefficient is positive.

To get the middle coefficient we need to make sure they are positive. That’s the only way they can ADD up to a positive 17. We need a 9x and an 8x. We can multiply two positives together or two negatives together to get positives.

Here’s the answer:
(-2x + 9)(x – 4) We can use trial and error by multiplying all the factors and adding ‘em up.
-2x2, 8x, 9x, -36.

-2x2 + 17x – 36
(-2x + 9)(x – 4)

Well done!

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Dividing Decimals: How to Turn a Decimal Into A Fraction

How to Turn a Decimal Into A Fraction

Three blind mice. Three blind mice.

See how they run. See how they run.

The Farmer’s Wife went after them with a carving knife

And divided them up for stew that night.

Filling eleven pots of rice.

Okay, that might not be the loveliest example of dividing decimals and converting into fractions and vice versa, but turning decimals into fractions is a useful skill to have. Plus, it’s super easy!

turn a decimal into a fraction

Dividing Decimals Example # 1.

Let’s take any decimal: 0.4

We know that 0.65 is more than half (0.5) but less than three quarters (0.75). But what is the exact fraction?

Let’s place the number over 1 in a fraction:
0.65/1

Then, count the number of digits in the decimal. We have 2 That means we must multiply it by that many place holders.

So we multiply both sides by 100
(0.65/1) X (100/100)

This gives you your fraction!
65/100

From there, we can reduce it to the simplest number possible:
13/20

Your answer is 13/20

Dividing Decimals Example # 2.

Let’s take a more complex number:

0.525

Set it up:

(0.525/1) X (1000/1000) = 525/1000

Then simplify it!

(525/1000) ÷ (5/5) = (105/200) ÷ (5/5) = 21/40

Your answer is 21/40

Dividing Decimals Example # 3.

Let’s take a very complex number:

0.54587123658654

Don’t be intimidated. This is a piece of cake!

0.54587123658654/1

Count the number of digits: 14

Multiply it by that number of place holders:

100000000000000/100000000000000

Then reduce the number by dividing both sides by 2…

54587123658654/100000000000000 = 27293561829327/50000000000000

27293561829327/50000000000000 is the answer!

Dividing Decimals Example # 4.

What if you have a mixed number?  Like, 5.623?

First, ignore the whole number. Set 5 aside.

Then, work with the decimal. 0.623

You can do it! Remember to count the number of digits after the decimal?

(623/1) X (1000/1000) = 623/1000

dividing decimals

Let’s go back to the Farmer’s Wife.

She had 3 mice and eleven cups of stew

3 ÷ 11 = 0.273

What is the fraction of mouse per stew?

273/1000

The number can’t be reduced further. This is the fraction of mouse per cup of stew!

Time for lunch!

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What is a Linear Function? How to Solve a Linear Function?

how to solve linear functions

If you’ve previously learned how to solve linear equations, you’re ready to move onto solving linear functions. They’re very similar, and the process also uses easy steps, so if you’re solving linear equations in your sleep, functions will be a breeze!

Step 1: Determine that you are, in fact, looking at a linear function.

In case you didn’t already know, a linear function will always begin with f(x) =, followed by the rest of the equation. It’s easy enough to remember, since the f stands for function. A linear function often closely follows the slope intercept formula you learned earlier in Algebra, which is y = mx + b.

slope intercept_how to solve linear functions

Step 2: Substitute variables for values as instructed.

When you’re first introduced to linear functions, you’ll most likely be given a value to substitute for your variable. Other functions will ask you to find the equation of a line, and then solve the corresponding function. In either case, substitute the values you’re given to solve for your variable.

Step 3: Solve for your variable and check your work.

Solve the function just like you would a linear equation; isolate your variable, do everything you did to one side of the equation to the other, and simplify your terms to their smallest values. You can check your work by solving the function with that value you’ve gotten. Just like with a linear equation, your function should look exactly how you found it if you’ve solved it correctly.

If this process isn’t as easy as you thought it would be, take a step back and refresh your memory on how to solve linear equations. It’s totally okay if you need more practice. Once you’ve mastered linear equations, solving linear functions will be the easiest thing you’ve ever done!

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How to Solve Linear Equations: It’s as Easy as 1,2,3!

how to solve linear equations_step by step

Linear equations are one of the first concepts you’ll be introduced to in Algebra I. If you’re like me, and you’ve struggled a good bit in your math journey, you may not find solving them very enjoyable or practical. However, as a longtime math survivor I’m here to tell you that’s absolutely not the case; engineers, architects, and surveyors of all kinds solve linear equations almost every day, and once you get the method down, you’ll forget why you had trouble in the first place. Linear equations can have one variable or many, but the method for solving linear equations of all types falls into three easy steps.

Solve Linear Equations Step 1: Isolate the variable you’re instructed to solve for.

how to solve linear equations_understand solutions to linear equations

If you’re faced with an equation that contains multiple variables from t to z, fear not; you’ll always start by trying to find the value of just one of them. Isolate your chosen variable on one side of the equation by going through your basic mathematic functions: add, subtract, multiply, and divide. You might use all four or just one, as long as your variable remains by itself on one side.

Solve Linear Equations Step 2: Everything you do to one side must be done to the other.

In order to keep your equation balanced and accurate, any functions you’ve done on one side of the equation must be done to the other. Remember that a positive number becomes negative when crossing to the other side, and the same goes for negative numbers. A misplaced positive or negative sign can really create chaos in your equation.

Solve Linear Equations Step 3: Simplify your terms and check your work.

how to solve linear equations

If you have any like terms, combine them. If the variable you’re solving for is no longer singular, divide it by itself until it is. When you think you’ve solved for your desired variable, use its determined value to work back through the equation to make sure you’ve solved it correctly. If you have, you’ll end up with the same equation you started with.

 

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What to Expect from College Algebra and Why this Subject Actually Matters?

college algebra_math homework help

When I was a little kid, math was just numbers. Add, subtract, multiply, and divide. Easy.

Then they started adding decimals. That was okay. I could divide a candy bar or an apple in half or thirds. Coins were decimals of dollars. It still made sense.

Then they added letters. Then things that were not letters or numbers. Like  the all-time favorite, Alebra 1

Like, whatever.

At that point, I gave up. Who cares and when will I ever need to learn this junk? Why study algebra in college? I was going to be a writer. Who needed Algebra 1, or the “New Math” version of Algebra? Like, when will I ever need to master a polynomial function? I deal with hyperbole, not hyperbolas.

It turns out that I did.

Let’s backtrack and look at the other side for a bit. Some critics say that algebra is holding back students. Kids are likely to drop out because they can’t pass proficiency exams, and algebra is the main reason why. Another article asks if algebra can help you understand the federal budget.

The answer is yes, it can.

Let’s address both points at once.

Algebra might not be completely applicable in life. Just like biology, or history, or poetry. But algebra does two things that are powerful and wonderful.

College Algebra challenges students to expand their minds.

college algebra_why it matters

Yeah, expansion can hurt. But like history or science, algebra introduces students to new ideas that they never considered. Algebra allows students to see the world in new ways—which may turn some off, but others who like seeing how things fit together in abstract ways are amazed at how the relationships between numbers change the world around them. Algebra is nothing if not an exercise in brain power.

College Algebra introduces life skills.

Do you need to know how to find a slope of a line? (Sure, you do! See my other blog! ). Or how about simply graphing X and Y? Plotting points is a basic tool in statistics to find out trends, conduct surveys, and understand data. If you’re going into marketing, finance, or anything that requires a chart, yeah, you need algebraic skills.

college algebra_math help

Not every skill is super useful. Quadratic formulas probably aren’t for everyone. But solving a quadratic formula involves breaking problems down into simpler puzzles, one step at a time. You end up dissecting a problem by looking at the sum of its parts. Want to see if a refinanced mortgage works for you? Check it out and figure it out, one step at a time.

College Algebra 1 - Math Help
And, yes, even as a writer, I needed algebraic skills. Algebra teaches students how to notice patterns in speech, description, and plotting. Poets count the number of beats and quantify them, and math can even inspire a story or two (Alice in Wonderland, anyone?).

So what should you expect from college algebra? A whole new world that is challenging, exciting, frustrating, and even beautiful in its aesthetics.

What do you get out of it? Depends on what you’re looking for. If it’s an equation, you get out what you get in. But the = sign suggests you get more than the sum of the parts.

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How to Solve a Quadratic Equation with the Quadratic Formula

what is the quadratic formula

Even mathematicians admit that the quadratic formula can look messy and overwhelming. However, try not to allow the multitude of letters, numbers and signs distract you from the importance of this very critical equation. Not only is it highly useful to you as a student studying math topics, but it has relevance in other subject areas, including physics and engineering.

Rather than see the quadratic equation as an overwhelming problem, let’s break it down step by step.

What the Quadratic Formula Means

When you see c, you are looking at a quadratic. The “x” corresponds with a specific line on a graph, usually the horizontal line. The vertical line of the graph, or “y” is understood as the “0”. The “a”, “b”, and “c” elements are simply place marks for numbers.

what is the quadratic formula. how to use it

A quadratic always gives you two answers for “x” so you can quickly see where they fall on the graph when “y” is zero. Some quadratics are very clean; you may even be able to work them out in your head.

For instance, x^2 – 3x – 4= 0 works out to (x – 4) x (x + 1) = 0. This means your two “x” answers are “4” and “-1”. Knowing this, you could plot those places on a graph and connect them with a curving line called a parabola. The high point, or vertex, of the parabola will correspond with the “x” line at -b/2a. For our sample above, with “b” as “-3” and “a” as “-4”, the result is 3/-8 or, in decimals, “-.375”.

We can plug “-3/8” into the original equation and find that y = (-3/8)^2 + 9/8 – 4.The answer comes to “-2.75”. Thus, we know the vertex of our quadratic equation’s corresponding parabola is at coordinates “-.375, -2.75.”

What Happens When Quadratics Are Complicated?

Our sample quadratics and corresponding parabola was relatively uncomplicated. However, when you have quadratics that don’t come together easily, you must use the quadratic equation.

The quadratic formula looks quite complex but is just another way of looking at ax^2 + bx + c = 0. Consequently, it involves solving a top half, -b +/- the square root of b^2 – 4ac, and then dividing the top half by 2a. This will give you the answer to “x” just as well as ax^2 + bx + c = 0.

We’ll look at the problem above to make sure it works:

quadratic formula

Our quadratic: x^2 – 3x – 4= 0

“-b” equals “3”

“b^2” equals “9”

“-4ac” equals “16”

“2a” equals “2”

Consequently, our answer is x = 3 +/- the square root of 25 divided by 2, or x = 4 and x = -1. These are the exact solutions we found at the beginning of our journey.

Give Yourself Time to Understand Quadratic Formulas and Equations

For most people, mastering quadratics is a process that requires a lot of time and energy. Plan to work out problems on your own time, and you’ll start to get faster and more accurate. Then, you can begin to apply the quadratic formula to plenty of real-life scenarios, from the height of a baseball tossed into the air at a particular velocity, to the time it takes a missile launched at a certain rate to hit its target.

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