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Algebra II – Study Guide

Polynomials

A monomial expression consists of a single variable with a whole and positive power.

Example: $y=x^2$

A polynomial expression consists of the sum of one or more monomials.

Example: $y=x^{2}+x+5$

When asked to find a root, it is another way of asking to find the solution to the expression set equal to zero.

Example: Find the roots of $x^{2}+5x+6\Rightarrow$ Factor to $(x+2)(x+3)=0$
So the roots are: -2, -3

The degree of a polynomial is the degree of the highest exponent of any variable in the equation.

Example: $x^{4}+4x^{2}-3=y$
Degree : 4 (because x is to the $4^{th}$ power, which is the highest exponent)

Example: $x^{2}+2x^{2}+5=y$
Degree : 2 (because x is to the $2^{nd}$ power, which is the highest exponent)

The Fundamental Theorem of Algebra says that every polynomial equation with a degree greater than zero has at least one root in the set of complex numbers.

Types of Algebraic Expressions

Linear Equations

Linear equations are basic polynomials with a degree of one. When plotted on a coordinate system, they will always form a line.

Example: Plot the following function: $y=2x+4$

Inequalities:

Inequalities contain four different signs (instead of the equal sign) and define the relationship between each side of the inequality.

Signs:

< Less than: the left side is smaller than the right side
> Greater than: the left side is larger than the right side
≤ Less than or equal to: the left side is smaller than or equal to the right side
≥ Greater than or equal to: the left side is larger than or equal to the right side

Example: $-3<-2x+7$

Subtract 7 from each side → $-10<-2x$
Divide each side by -2 → $-5>x\ (Solution)$
Plot:

Remember: Inequalities are very similar to equations except that when you are multiplying or dividing each side of the inequality by a negative number, the inequality sign is reversed.

Logarithmic Expression:

Definition of a logarithm: $If\ y=b^{x},\ then log_{b}y=x$

The exponent x in the exponential expression $b^x$ is the logarithm in the equation $log_{b}y=x$ . The base b in $b^x$ is the same as the base b in the logarithm. A positive number b, raised to any power x, cannot equal a number y less than or equal to zero. Therefore, the logarithm of a negative number or zero is undefined (that means that there is no solution).

Example: Evaluate $log_{8}16$

Step 1: Write an equation in logarithmic form → $log_{8}16=x$
Step 2: Convert to exponential form → $16=8^x$
Step 3: write each side using base 2 → $2^{4}=(2^{3})^x$
Step 4: Power Property of Exponents → $2^{4}=2^{3x}$
Step 5: Set the exponents equal to each other → $4=3x$
Step 6: $\frac{4}{3}=x$ , so the answer of $log_{x}16=\frac{4}{3}$

Graphing logarithmic functions

Logarithmic functions look like inverse functions of exponential functions.

Summary : Translations of Logarithmic Functions

Characteristic	$log_{b}x$	$y=log_{b}(x+h)+k$
Asymptote	x = 0	x – h = 0, or x = h
Domain	x > 0	x > h
Range	All real numbers	All real numbers

Exponential Equations

An equation in the form of $b^{cx}=a$ , where the exponent includes a variable.

Example: Solve $7^{3x}=20$

Step 1: Take the $log_{10}$ of each side ⇒ $log_{10}7^{3x}=log_{10}20$
Step 2: Use the power property of logarithms ⇒ $3xlog_{10}7=log_{10}20$
Step 3: Divide each side by $3log_{10}7$ ⇒ $x=\frac{log_{10}20}{3log_{10}7}$
Step 4: Evaluate (use a calculator) ⇒ $x=0.5132$

Exponential Growth and Decay

Exponential functions can model growth or decay. For example, $y=ab^x$ , when b > 1, models growth while 0 < b < 1 models exponential decay.

Inverse Functions

If $f$ and $f^{-1}$ are inverse function; then, $f(a)=b$ if and only if $f^{(-1)}(b)=a$

Example: Find the inverse of $f(x)=\frac{x+6}{2}$

Step 1: Replace f(x) with y in the original equation ⇒ $y=\frac{x+6}{2}$
Step 2: Interchange x and y ⇒ $x=\frac{y+6}{2}$
Step 3: Solve for y ⇒ $2x-6=y$
Step 4: Replace y with $f^{-1}(x)$ ⇒ $y=2x-6\rightarrow f^{-1}(x)=2x-6$
Answer: $f^{-1}(x)=2x-6$

Solving Radicals

A Radical Equation contains at least one variable expression that is found inside of a radical, usually a square root (but could also be a cubed root, fourth root, etc.)

Examples of radical equations:

$\sqrt{x}+10=4$
$(x-5)^{\frac{2}{3}}=30$

NOT a radical equation:

$\sqrt{10}+2x=10$

Example: Solve $3+\sqrt{3x-2}=7$

Step 1: Isolate the radical ⇒ $\sqrt{3x-2}=4$
Step 2: Square each side of the equation ⇒ $(\sqrt{3x-2})^{2}=4^{2}$
Step 3: Simplify the expressions ⇒ $3x-2=16$
Step 4: Add 2 to both sides ⇒ $3x=18$
Solution: $x=6$

Adding or Subtracting Radical Expressions

Be careful not to confuse radical expressions with radical equations. Radical equations must contain a variable inside the radical and utilize an equal sign. Expressions do not have to contain a variable inside the radical.

Example: solve $4\sqrt[3]{x}-3\sqrt[3]{x}$

Step 1: Combine like terms ⇒ $(4-3)\sqrt[3]{x}$
Step 2: Subtract ⇒ $(1)\sqrt[3]{x}$
Step 3: Simplify ⇒ $\sqrt[3]{x}$

Power Functions

Power functions come in a standard form $y=ax^b$ . They are very similar to exponent functions; however, in a power function the variable base is raised to a fixed exponent.

a: called the scaling factor. It moves the $x^b$ term up or down by a factor of a
b: determines the function’s overall shape

Practice: Graph $y=3x^{2},\ y=-3x^{2},\ y=3x^{-2},\ y=3x^{0},\ y=3x^{\frac{1}{2}},\ y=3x^{1}$

Dividing Polynomials

If you’re dividing a polynomial by something more complicated than just a simple monomial, then you’ll need to use a different method for the simplification.

Example: $x-5\overline{)x^{3}-3x^{2}-5x-25}$

To find the solution to this problem we have a few methods to try. The first is long division. The second is synthetic division. Synthetic division is easier and takes less space.

For synthetic division the first step is to find the root of the divisor.

$x-5=0,\ x=5$

Then you set up a series of placeholders

Problem Coefficients: $x^{3}\ x^{2}\ x^{1}\ |\ x^{0}$
Answer Coefficients: $x^{2}\ x^{1}\ x^{0}\ | R$

5| 1 – 3 – 5 | -25

– – – | –
_______________________________________

– – – | –

Add down the column, then multiply by the root and place it in the next column. Repeat.

Problem Coefficients: $x^{3}\ x^{2}\ x^{1}\ |\ x^{0}$

Answer Coefficients: $x^{2}\ x^{1}\ x^{0}\ | R$

5| 1 – 3 – 5 | -25

0 5 10 | 25
_______________________________________

1 2 5 | 0

Now all that is left is use the answer placeholders to convert the coefficients back into an algebraic expression.

$x^{2}+2x+5$

$x-2\overline{)x^{2}+3x-10}$

Take the highest powered term from the divisor and the highest powered term from the dividend only to get the first term in the answer.

Just like normal long division, once you have the first digit, or in this case, term, you multiply by the divisor and subtract from the dividend.

To get the next term in the answer, you take the highest powered term in the new dividend and divide by the highest powered term in the divisor.

So now that all is divided evenly, you have your solution, $x+5$ .

Example: $x-5\overline{)x^{3}-3x^{2}-5x-25}$

Take the highest powered term from the divisor and the highest powered term from the dividend to get the only first term in the answer.

Just like normal long division, once you have the first digit, or in this case, term, you multiply by the divisor and subtract from the dividend.

To get the next term in the answer you have the highest powered term in the new dividend and divide by the highest powered term in the divisor again.

To get the next term in the answer, you take the highest powered term in the new dividend and divide by the highest powered term in the divisor

$x\overline{)5x}=5$

So now that it all divided evenly you have your solution, $x^{2}+2x+5$ .

Completing the Square

Completing the square can always be done in 6 steps:

Step 1: Divide the coefficient on the $2^{nd}$ term by 2.
Step 2: Take the square root of the first term.
Step 3: Add steps 1 and 2; then, square term.
Step 4: Use FOIL to expand the binomial.
Step 5: Subtract from the original
Step 6: Put together step 3 and 5

Example: $x^{2}+14x+27$

Step 1: Divide the coefficient on the $2^{nd}$ term by 2.

$\frac{14}{2}=7$

Step 2: Take the square root of the first term.

$\sqrt{x^{2}}=x$

Step 3: Add steps 1 and 2; then, square term.

$(x+7)^{2}$

Step 4: Use FOIL to expand the binomial.

$x^{2}+14x+49$

Step 5: Subtract from the original

$\ x^{2}+14x+27$
$-x^{2}-14x-49$
__________________
$0+\ \ 0\ \ -22$

Step 6: Put together step 3 and 5

$(x+7)^{2}-22=x^{2}+14x+27$

Example: $x^{4}+8x^{2}$

Step 1: Divide the coefficient on the $2^{nd}$ term by 2.

$\frac{8}{2}=4$

Step 2: Take the square root of the first term.

$\sqrt{x^{4}}=x^{2}$

Step 3: Add steps 1 and 2; then, square term.

$(x^{2}+4)^{2}$

Step 4: Use FOIL to expand the binomial.

$x^{4}+8x^{2}+16$

Step 5: Subtract from the original

$\ x^{4}+8x^{2}$
$-x^{4}-8x^{2}-16$
__________________
$0+\ \ 0\ \ -16$

Step 6: Put together step 3 and 5

$(x^{2}+4)^{2}-16=x^{4}+8x^{2}$

Factoring

The first thing to check when you are factoring is if there is a greatest common factor (GCF) that is not 1. IF there is a GCF > 1, you should divide every term by the GCF to factor it out front.

Example:

$3x^{2}+12x+12$
Divide by the common factor, 3
$=3(x^{2}+4x+4)$

Example:

$5x+7x+5y+7y$
Group by a common factor, 5 and 7
$-[5(x+y)]+[7(x+y)]$
Then, group again because (x+y) is repeated
$(x+y)(5+7)$

Difference of Squares

$x^{3}-y^{3}=(x-y)(x^{2}+xy+y^{2})$

The difference of two squares formula is critical and you should be able to recognize it.

Example: $x^{2}-y^{2}=(x+y)(x-y) Perfect Squares formulas$ x^{2}+2xy+y^{2}=(x+y)^{2}x^{2}-2xy+y^{2}=(x-y)^{2} $General trinomial$ acx^{2}+adx+bcx+bd=(ax+b)(cx+d)

Example:

$3x^{2}+10x+8$

You need to find factors (x )(3x ) so that the product of the two missing numbers is 8.

Since 8 can be written as the product of 2 and 4, and since 2 + 4 = 8, two and eight are a reasonable choice to complete the factors

$(x+2)(3x+4)$

Use the FOIL method to verify that the answer is correct.

Composite Functions

Composite functions are combinations of functions that result in a different result than either of the original functions. There are several different composite operations as shown below.

Example:

$f(x)=a^{3},\ g(x)=a^{2},\ find\ fog$
$fog=f(g(x))=f(a^{2})=(a^{2})^{3}=a^{6}$

Example:

$f(x)=a^{3},\ g(x)=a^{2},\ find\ g(f(x))$
$g(f(x))=g(a^{3})=(a^{3})^{2}=a^{6}$

Example:

$f(x)=a^{3},\ g(x)=a^{2},\ find\ f\cdot g$
$f\cdot g=f(x)g(x)=a^{3}\cdot a^{2}=a^{5}$

The Quadratic Formula

The general form for a quadratic equation is:

$ax^{2}+bx+c=0:\ where\ a\neq 0$

The Quadratic Formula can be used to find the root(s) of any quadratic equation:

$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

The discriminate is the expression under the square root and can be used to predict the roots:

$if\ b^{2}-4ac>0\rightarrow there\ are\ two\ real\ roots$
$if\ b^{2}-4ac=0\rightarrow there\ is\ a\ real\ repeated\ root$
$if\ b^{2}-4ac<0\rightarrow there\ are\ two\ complex\ roots$

Example: $Solve\ 3x^{2}+2x-4=0,\ for\ x$

Recognize that a = 3, b = 2, and c = -4 (from the coefficients)
Plug these numbers (a, b, and c) into the quadratic formula:
$x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{-2\pm\sqrt{2^{2}-4(3)(-4)}}{2(3)}=\frac{-2\pm\sqrt{52}}{6}=\frac{-2\pm2\sqrt{13}}{6}=\frac{-1\pm\sqrt{13}}{3}$

Conic Sections

Parabolas, circles, and ellipses, and hyperbolas are considered conic sections because they are formed when a cone or double cone is sliced by a plane.

Standard Equation

The general form of any conic section, where A, B, and C are not all zero is
$Ax^{y}+Bxy+Cy^{2}+Dx+Ey+F=0$

Standard Form of Equations

Parabola Equation: $y=a(x-h)^{2}+k\ or\ x=a(y-k)^{2}+h$

Circle: $(x-h)^{2}+(y-k)^{2}=r^{2}$

Ellipse: $\frac{(s-h)^{2}}{a^2}+\frac{(y-k)^2}{b^2}=1$

Hyperbola: $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$

Circle

Set of all points in a plane that are an equal distance from a center point.

$(x-h)^{2}+(y-k)^{2}=r^{2}$ with center: (h, k) and radius: r

Parabola

Set of all points in a plane that are an equal distance from a given point and line.

$y=a(x-h)^{2}+k\ or\ x=a(y-k)^{2}+h$

Vertex: (h, k) and Axis of symmetry: x = h

“a” in the equation tells which way the parabola will open

Negative a: opens downward
Positive a: opens upward

Directrix: the distance from any point to the focus must be the same as the distance from that point to the directrix.

Hyperbola

Equations for a Hyperbola with the Center at the Origin

Standard form of Equation	$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$	$\frac{(y-k)^2}{b^2}-\frac{(x-h)^2}{a^2}$
Direction of Transverse Axis (TA)	Horizontal	Vertical
Foci	(c, 0), (-c, 0)	(0, c), (0, -c)
Vertices	(a, 0), (-a, 0)	(0, a), (0, -a)
Length of TA	2a	2a
Length of Conjugate Axis	2b	2b
Equations of Asymptotes	$y=\pm\frac{b}{a}x$	$y=\pm\frac{a}{b}x$

Example: Write $x^{2}+2y^{2}-2x-8=0$ in standard form. Then, state which conic section the graph of the equation would describe.

Step 1: Rewrite the original equation ⇒ $x^{2}+2y^{2}-2x-8=0$
Step 2: Isolate terms ⇒ $x^{2}-2x+$ ___ $+2y^{2}=8+$ ___
Step 3: Complete the square ⇒ $x^{2}-2x+1+2y^{2}=8+1$
Step 4: Condense the equation ⇒ $(x-1)^{2}+2y^{2}=9$
Step 5: Divide each side by 9 ⇒ $\frac{(x-1)^2}{9}+\frac{2y^2}{9}=1$
Step 6: Graph the equation (an ellipse)

Rationalizing Denominators

A denominator needs to be rationalized if it contains a square root or a root of any kind. To get rid of the square root in the denominator, multiply both the top and bottom of the expression by the square root (the same as multiplying by 1). The square root on the bottom will cancel out.

Example: Rationalize $\frac{1}{\sqrt{3}}$

$\frac{1}{\sqrt{3}}=\frac{1}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{\sqrt{3}\cdot\sqrt{3}}=\frac{\sqrt{3}}{\sqrt{9}}=\frac{\sqrt{3}}{3}$

Example 2: Rationalize $\frac{\sqrt{108}}{3\sqrt{2}}$

$\frac{\sqrt{108}}{3\sqrt{2}}=\frac{\sqrt{108}}{3\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{108}\cdot\sqrt{2}}{3\sqrt{2}\cdot\sqrt{2}}=\frac{\sqrt{216}}{3\sqrt{4}}=\frac{\sqrt{216}}{6}$

To simplify 216, make a factor tree and determine what its factors are.

So we pull out a 2 and a 3 and are left with a 2 and a 3 inside.

$\frac{2\cdot 3\cdot \sqrt{2\cdot 3}}{6}=\frac{6\sqrt{6}}{6}=\sqrt{6}$

Absolute Value

The absolute value of a number is the distance that number is from zero. Look at the number like below. The blue dots represent -3 and 3. Both dots are the same distance, 3, from zero on the number line. So, the absolute value of 3 and the absolute value of -3 are both zero.

Examples:

$|-1|=1$
$|2|=2$
$|0|=0$

This means that when you evaluate a variable (like x) in an equation, you do not know if it is positive or negative. Normally the answer is x equals plus or minus some value.

$|x|=1,\ x=\pm 1$

If it is not just the absolute value of x, you will still have two solutions, but they will not be opposites of each other.

$|x+1|=2,\ x+1=\pm 2,\ x=-3\ or\ 1$

The graphs of absolute value functions always look like a “V”. the “V” is wider if there is a fraction in front of the variable in the equation and narrower if there is a whole number coefficient in front of the variable. Here are some examples of how the absolute value graph changes based on transformations to the equation: