Algebra I

The slope is the slant of a line.  All diagonal lines have a slope. Horizontal lines have a slope of 0. Vertical lines have an undefined slope. 

Lines that slant upward, towards Quadrant I in the coordinate plane (where x and y are both positive), have positive slopes.

Remember, every linear equation forms a line.  Each line contains a set of points whose coordinates are given as $(x, y)$.  The coordinates $(x,y)$ not only show the points on the line, but they provide the $x$ and $y$ variables in the linear equations (e.g., in $y = mx + b$).

If you are given a coordinate (or both coordinates) of a point on a line, you can plug those coordinates into the x and y of the linear equation. 

There are many ways to write a linear equation (an equation that represents a line).  When we graph a line, we like to put it in $y=mx+b$ form, because that makes it easy to find the slope and the y-intercept.  But, lines can also be given in standard form. 

Standard form is: 

$Ax+By=C$

Essentially the $x$ term comes first, added to the $y$ term, equal to the number (which we call $C$).

Fractions show parts of wholes. The numerator tells how many pieces a whole is cut into and the numerator tells how many of those pieces you have.

Because rational expressions are just fractions with variables, the process of dividing rational expressions is exactly the same as the process of dividing fractions. You just find the reciprocal of the second rational expression (flip it, so the the denominator becomes the numerator and the numerator becomes the denominator), and then multiply the rational expressions (numerator times numerator and denominator times denominator).  It's really that easy!

Because rational expressions are just fractions with variables, the process of multiplying rational expressions is exactly the same as the process of multiplying fractions. Multiply the rational expressions just like you multiply fractions (numerator times numerator and denominator times denominator).  

Rational expressions are fractions that have variables (including binomals or polynomials) in them.  The variables can be in the numerator or the denominator (or both!).

The rules for adding and subtracting rational expressions are exactly like those for adding and subtracting fractions.  If the denominators are the same, you can add or subtract the numerators.  If the denominators are different, you have to find a common denominator before you can add or subtract the numerators. 

Division with polynomials looks complicated.  But you can do long division with polynomials in much the same way that you do division with numbers!  And, like with long division, you just have to pay attention to where you put your answer terms so that you can keep track of where you are in the problem. 

To do polynomial division, you use the long division "house."  Let's review that process very quickly:

The problem: $56 \div 4=$ is written: $4\overline{)56}$

When you multiply polynomials, you must make sure that you multiply each term by each other term.  When multiplying binomials, we ensure that each term is multiplied by each other term using the FOIL process.

Foil is an acronym that stands for first, outer, inner, last.

The basic process of using foil, is to take the two binomials and first multiply the "first" terms of each binomial together.

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Sometimes algebra problems will ask you to combine polynomials with multiplication.  These problems can look intimidating, but they are very similar to multiplying binomials (remember FOIL?).  The secret to these problems is being methodical and making sure that every term in each polynomial is multiplied by every term in each of the other polynomials. 

Remember that: