## Simplifying polynomials

“Polynomials” are expressions with two or more (poly) types of numbers (nomials). An example of a polynomial expression would be:

$(6x^2+5x+3)+(x^2+3x-2)$

To simplify this expression, we need to first consider PEMDAS. Within each set of parentheses, nothing can be reduced or combined. Also, no exponents can be computed since we don’t know what x stands for—it’s already reduced as far as we take reduce it. There is no multiplication or division. There’s only addition and subtraction. So, we must now “combine like terms.” Like terms are the simply the ones “like each other.” You can ask, “How many *x-squareds* are there?” Well, there are 6 *x-squares* plus 1 more *x-squared*, which makes 7 *x-squared*. “How many x’s are there?” There are $5x$’s plus $3x$’s, to make $8x$’s. And looking at the numbers only, 3-2 is 1. So, our simplified polynomial is:

$7x^2+8x+1$

## Factoring binomials using F.O.I.L.

Factoring binomials means to multiply binomials (expressions with two types of numbers). For example, this type of problem might be:

$(x+3)(x-1)$

This type of problem is solved through the “F.O.I.L.” method. F.O.I.L. stands for “First, Outer, Inside, and Last.” Using the problem above, let’s F.O.I.L.:

- First: $x \times x$ yields $x^2$
- Outer: $x \times -1$ yields $-1x$ or just $-x$
- Inner: $3 \times x$ yields $3x$
- Last: $3 \times -1$ yields $-3$

$x^2-x+3x-3$

Lastly, we combine like terms. The only terms able to be combined are the $-x$ and the $3x$ which gives us $2x$. So, our final answer is:

$x^2+2x-3$

## Factoring trinomials

This implies doing just the opposite of what we just did. It’s like taking the expression $x^2+2x-3$ and working backwards to find the factors: $(x+3)(x-1)$

Consider this example for factoring trinomials:

$x^2-9x+20$

To find the factors for this trinomial:

- Find factors for $x^2$ : $x$ times $x$ or $(x)(x)$
- Find factors for 20: $1 \times 20$, $2 \times 10$, and $4 \times 5$
- Find the pair with a sum of -9. -4 and -5 have a sum of -9
- Complete the factors: $(x-4)(x-5)$
- Double check using F.O.I.L. Be sure the numbers are correct and the positives and negatives are correct. Using F.O.I.L., you’ll get back to $x^2-9x+20$ and know your answer is correct.

## Dividing Polynomials

Dividing polynomials can appear intimidating. Don’t get psyched out! The basic process is the same as regular old long division that you did in grade school. Consider the division problem of…

It’s a binomial (x + 1) dividing into a polynomial (the parts inside the division sign). Look at the graphic following this sentence for an explanation to solve the problem.

## Handling square roots

It’s necessary to be able to simplify, multiply and divide square roots.

Simplifying square roots:

Suppose you have an answer of $\sqrt {2}$, you’d need to simplify it on down to its most basic form. To simplify square roots, think of any perfect squares that are factors of the number inside the square root symbol. In other words, do 4, 9, 25, 36 etc. divide evenly into the number inside the symbol? If so, it needs to be simplified. In this case, it could’ve been written $\sqrt{9 \times 2}$ So, we need to “pull out” the 9. The square root of 9 is 3, so our final and most simplified answer would be $3 \sqrt{2}$

Multiplying square roots:

Multiplying square roots is simple—you multiply the numbers under the square root sign, then look to simplify. Consider the problem: $\sqrt{2} \times \sqrt{8}$. Just multiply $2 \times 8$ to get $\sqrt {16}$ The square root of 16 is 4, the answer.

Dividing square roots:

Consider the problem: $\sqrt{27} \over \sqrt{12}$

First simplify by “pulling out” the 9 in the numerator and the 4 in the denominator. The problem then becomes: $3 \sqrt{3} \over 2\sqrt{3}$ Now simplify. The two $\sqrt{3}$ factors cancel each other out, so you could just scratch them and you’re left with your answer: $3/2$

## Distance between two points in a plane

Algebra is often called “linear algebra” because the equations can be graphed on a plane, that is to say, on graph paper. This skill requires you to, when given to points on a plane, calculate the distance between those two points.

There are two methods of finding the distance between two points in a plane:

1. Graphically – you can draw the graph to visualize it.

2. Formulaically – you can use the distance formula shown below.

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

We’ll use both methods to illustrate how to compute the distance.

1. Graphically solving the problem:

Consider the example: Find the distance between point A at (-2, 6) and point B at (5,3). Refer to the graphic below to solve this problem.

2. Formulaically solving the problem:

The formula for computing the distance between two points on a graph is built on the same principles that we used in the problem above. The formula is:

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Consider the same two points that we used up above: point A (-2,6) and point B (5,3). Placing the x and y coordinates into the formula gives us:

$d=\sqrt{(5- -2)^2+(3-6)^2}$

Which becomes…

$d=\sqrt{7^2+(-3)^2}$

Which becomes…

$d=\sqrt{49+9}$

Which becomes…

$d=\sqrt{58}$ or $d\approx 7.62$

## Ratios or proportions

Ratios or proportions are essentially the same thing. They show how numbers relate to one another. Ratios and proportions can be shown a couple of ways:

- In numbers such as 3:1. This says “three to one.”
- As fractions such as $3 \over 1$ This fraction is the same as writing 3:1. It is read as, "A ratio of 3 to 1."
- Or in words such as there are 3 teaspoons to every 1 tablespoon. It’s still “three to one.”

Often, questions on standardized tests have you solve a simple proportion problem. It may go something like this:

It took 8.5 bags of mulch to cover a playground with 500 square feet of area. How many bags of mulch would it take to cover an 800 square foot playground?

This may appear intimidating, but it’s very easy. It’s really saying 8.5 is to 500, as what is to 800? First, let “what is” be $x$ Therefore, to define the variable, $x$ = the number of bags to cover 800 square feet. Next, we put the problem in the form of a proportion—that is to say, we put it in fraction form and set them equal to one another. So, we now have:

$8.5 \over500$ = $x \over 800$

This now is read as: “8.5 is to x, as 500 is to 800.”

To solve a proportion like this, we cross-multiply and set each “line of the cross” equal to each other. So $8.5(800)=500x$ This gives us $6800=500x$ Divide both sides by 500 to get the answer, 13.6 bags.

- As a note, on a question like this, the answer will likely be “14 bags” not 13.6 because you can’t buy a “.6” bag.