VIDEOS for INTERMEDIATE ALGEBRA

Listed below are descriptions of the most common topics in our Intermediate Algebra course along with links to videos showing you examples of how perform various task.

Algebra of Functions

Rules of Exponents Part 1 (09:24) - The video is the first of two videos on the rules for exponents. This video discusses the addition, power and quotient rule.

Rules of Exponents Part 2 (11:34) - This video discusses the zero power rule, negative exponents and more on the power rule.

Adding Polynomials (06:06) - Like terms are terms that have the exact same variables raised to the exact same powers. For example, 4x²y⁶z and -6x²y⁶z are “like” terms. When adding, combine like terms.

Subtracting Polynomials (08:49) - When subtracting, remember to DISTRIBUTE the minus sign to EACH TERM being subtracted. Then, combine like terms. For example, (4x - 3) - (2x - 8) equals 4x - 3 - 2x + 8. Notice the minus sign was distributed to both the 2x and the -8. Then, combining like terms, you get 2x + 5 as a final answer.

Multiplication of Polynomials (10:58) - When multiplying, be careful. Make sure you multiply each term of the first polynomial by each term in the second polynomial.

Composition of Functions (08:58) - A big piece of advice here is to NOT confuse composition of functions with multiplication of functions!!! To determine f(g(x)), you need to look at the function f(x) and EVERY place you find an x, replace it with the entire function g(x). Remember that if x was squared in f(x), then you must square g(x) in the composite function.

Domain of a Function (06:10) - The domain of a function are the variables that the independent variable (frequently called "x") can take on and produce a valid number for f(x). For example, the domain of any polynomial is all real numbers.

The domain of a rational expression is all real numbers EXCEPT where the denominator would be 0. For example, the domain of

f(x) = (x + 2) divided by the quantity ( (x - 3) * (x + 5) )

where the denominator is (x -3)(x + 5) would be all real numbers EXCEPT 3 and -5 because both 3 and -5 cause the denominator to equal 0 and we can never divide by 0.

Factoring

Factoring – There are several different types of factoring that you will be responsible for during the course of the semester. The key to remember here is always start by taking out the GCF. Then you can use either trial-and-error or the “AC Method” of factoring.

Taking out the GCF (02:19) - When you are looking to take out the greatest common factor, first look at the coefficients and see if there is a common factor. Then look at the variables. If a variable is present in EACH term, then look at the exponents. Finding the smallest exponent and that is what you will take out.

If you have an expression like 12x⁵ – 32x³-16x², you can factor out the greatest common factor of 4 since 4 divides evenly into 12, 32 and 16. Looking at the variables, there is an “x” in each term. Since the smallest exponent is 2, we can factor out an x².

Therefore, 12x⁵ – 32x³-16x²factors to 4x²(3x² – 8x – 4).

There may be more than one variable that is common to each term in a polynomial. The video Factoring GCF – 2 Variables (02:11) will show you how to factor 2 variables from a polynomial.

Factoring By Grouping (02:30) – When you factor by grouping, have four terms. You will take the GCF out of the first two terms, then take the GCF out of the second two terms. Remember to keep the sign when you take out the GCF. What you are left with should be the same. Then take out the factor of what is common to both groups. For example, if you have

6x² – 8x – 9x + 12

The GCF for the first two terms is 2x. The GCF for the second two terms is -3. Taking out the GCF, you get 2x(3x – 4). Taking out the GCF for the second two terms, you have -3(3x – 4). Notice the common factor of (3x – 4). When we factor out (3x – 4), we get (3x – 4)(2x – 3), which is the final answer.

AC Method of Factoring (5:06) – The AC method is another way of factoring a quadratics like 6x² – 17x + 12. Here, you multiply AC and find the pair of factors that add up to B. Be careful with your signs.

AC = 6 * 12 = +72

The factors of 72 are (1, 72), (-1, -72), (2, 36), (-2, -36), (3, 24), (-3, -24), (4, 18), (-4, -18), (6, 12), (-6, -12), (8, 9) and (-8, -9).

The pair of factors that add up to -17, the coefficient of our B term is (-8, -9)

Replace the B term by the sum found in the last step. That gives us 6x² – 8x – 9x + 12.

Now factor by grouping. 2x(3x – 4) -3(3x – 4) which simplifies to (2x -3)(3x – 4).

Difference of Perfect Squares (01:47) : a² – b² = (a – b)(a + b)

Factoring – Sum of Perfect Cubes (02:31) : a³ + b³ = (a + b)(a² – ab + b²)

Factoring - Difference of Perfect Cubes (02:11) : a³ – b³ = (a – b)(a² + ab + b²)

Solving Polynomial Equations (06:44) : Here, you will be solving equations of the form x² + 5x + 6 = 0. First, you factor to (x + 2)(x + 3) = 0. Next, you set each factor equal to 0 and solve. For this example, if x + 2 = 0, then x = -2. If x + 3 = 0, then x = -3. That makes the solution to this equation x = -2 and x = -3.

Rational Expressions & Equations

Simplifying Rational Expressions (05:01) – To reduce a rational expression, you factor the numerator and factor the denominator. Then cancel out the common factors.

Multiplying Rational Expressions (11:44) - When multiplying rational expressions, you need to factor all the numerators and all of the denominators. Then cancel factors common to the numerator and denominator.

Dividing Rational Expressions (06:43) - When dividing rational expressions, first, factor all of the numerators and all of the denominators. Then take the second fraction (called the divisor), flip it, and change the question to multiplication. Then cancel out any common factors. Keep in mind this is the same procedure you would do when dividing ordinary fractions.

Adding Rational Expressions (06:08) - You will really need to apply those factoring skills to find the LCD of your rational expressions. However, once you have done that, all you need to do is create equivalent fractions and combine like terms in your numerator. At this point, you MIGHT have to again factor your numerator and then see if any factors from the numerator will cancel any factors in the denominator.

Subtracting Rational Expressions (07:12) - Again ffactoring skills are critical to find the LCD of your rational expressions. Once you have done that, distribute the minus sign to EACH TERM being subtracted. Then combine like terms and simplify if necessary.

Solving Rational Equations #1 (03:52) - The key to solving rational equations is to first find the LCD of the equation. Next, multiply each term by the LCD. You will notice the denominators all cancel. Then you just need to solve the resulting equation.

Solving Rational Equations #2 (Domain issue with "solution") (5:53)- Sometimes when you "solve" a rational equation, your "solution" really isn't a solution after all. That is because the "solution" causes one of the original expressions to have a zero in the denominator. This video will show you how this can happen.

Word Problems - Pythagorean Theorem (05:11) - The Pythagorean Theorem states that c² = a² + b². "c" is the length of the hypotenuse. "a" and "b" are the lengths of the other two sides.

Word Problems - Work Problems (03:16) - Work problems involve the time it takes two people to complete a task when working together. The general formula is

t/a + t/b = 1

where "t" is the total time it takes both people to do a job. "a" is the time it takes for one person to complete the job alone and "b" is the time it takes for the other person to complete the job on their own.

Radicals & Rational Exponents

Simplifying Radicals - When the Radicand is Perfect (05:21) - A square root is said to be simplified if there are no perfect squares under the square root symbol. A cube root is said to be simplified if there are no perfect cubes under the cube root symbol.

Simpfying Radicals - When the Radicand is NOT Perfect (09:38) - If the radicand is not perfect, you might still be able to simplify it. For example .

Rational Exponents (10:54) - Radicals are actually fractional exponents. For example, can be written as (50x⁷)^1/2

Multiplying Radicals - Like Index (02:09) - When you multiply radicals with the same index, you need to simply multiply the radicands (what is under the radical symbol). Then remember to simplify the radical if necessary.

More Multiplication of Radicals (08:44) - This is another video with more examples involving multiplication of radicals.

Rationalizing Denominators (07:22) - You need to create perfect powers in the denominator. For example, if you have a square root in the denominator, then you need to each item under the square root symbol is a perfect square.

Next, multiply the numerator by whatever you multiplied the denominator by.

Simplify and you are done.

More Rationalizing Denominators (05:02) - This is another video with more examples involving rationalizing denominators.

Adding and Subtracting Radicals (04:09) - You can only add or subtract "like" radicals. That means you must simplify each radical before you combine them. This video covers examples of both adding and subtracting radicals.

General Example (03:37) - This video shows another example combining many of the techniques discussed earlier.

Solving Radical Equations (07:13) - The key here is to first isolate the radical (make sure it is alone on one side of the equal sign). Then raise both sides to the same power.

For example,

Square both sides and you get x + 4 = 16. That means x = 12.

More Solving Radical Equations (05:15) - This video contains more examples of solving radical equations.

Complex Numbers & Quadratic Formula

Simplifying Powers of "i" (04:22) - This video will show you how to simplify the powers of i. What you need to remember is that i² = -1.

Adding and Subtracting Complex Numbers (04:11) - Addition of Complex Numbers requires you to combine the real parts and combine the imaginary parts. For example,

(4 + 5i) + (6 + 9i) = (4 + 6) + (5i + 9i) = 10 + 14i.

Subtraction of Complex Numbers requires you to distribute the minus sign BEFORE you combine like terms. For example

(1 + 2i) – (3 – 4i) = 1 + 2i – 3 – (-4i)

Then combine the real parts and combine the imaginary parts.

(1 – 3) +(-3i + 4i) = -2 + 1i

Multiplying Complex Numbers (08:46) - The procedure for the Multiplication of Complex Numbers is the same as for multiplying any other binomial by another binomial. You need to FOIL.

Division of Complex Numbers (07:52) – The Division of Complex Numbers requires you to multiply both the numerator and denominator by the complex conjugate of the denominator.

The Complex Conjugate of (a + bi) is (a – bi).

Equality of Complex Numbers (02:57) - Two Complex Numbers are equal if the real parts are equal AND the imaginary parts are equal. For example, if

a + 4i = -7 + bi

then a = -7 and b = 4.

Solving Quadratic Equations via the Square Root Method (12:36) - One method you can use to solve a quadratic equation is the square root method. To use the square root method

Put the constant alone on one side of the equal sign.
Complete the Squares – To complete the square, first set up your quadratic such that the constant is alone on the right side of the equal sign. Then take half of the value of the "b", square it, and add to both sides of the equal sign. You will need to know how to complete the squares in order to solve quadratic equations via the square root method.
Take the square root of both sides.
Remember to take both the positive and negative square root.

Quadratic Formula Part 1 (11:25) - The quadratic formula is x = ( - b + or - sqrt ( b^2 - 4*a*c)) divided by (2 * a) . This video will show you how to solve quadratic equations using the quadratic formula.

Quadratic formula on the TI 83 (04:35) - To enter the formula into the calculator, simply type (-b + sqrt(b^2 - 4*a*c)) / (2*a)

Make sure the numerator and denominator are both enclosed in parentheses. The parenthesis are REQUIRED here.

Understanding the Discriminant (10:47) - The Discriminant is the radicand (b² - 4ac) in the quadratic formula. This video will show you how the discriminant relates to the graph of a quadratic function.

Exponential Functions

Domain of a Logarithmic Function (03:21) - This video will show you how to find the domain of a logarithmic function. Simply put, the arguement of a log function must be greater than 0. To determine the domain of f(x) = log (x + 7), we need to solve the inequality x + 7 > 0. By subtracting seven from both sides, we get x > -7. That means the domain of f(x) = log (x + 7) is all real numbers greater than -7.

Converting Exponential to/from Logarithmic Form (04:32) - This video will show you how to convert a = bc to/from logb a = c. You'll learn about a memorization technique called "logs are used to Build A Cabin."

Banking Problems (08:57) - If you had $500 to invest in a CD for 4 years at 2.3%, how much money would be in the CD at the end of four years? This video will show you how money can grow in a savings account or CD.

Change of Base Formula (03:16) - Calculators can determine logs when the base is 10 or natural logs (ln). However, if you have to evaluate log₄ 5, you need to convert it to a quotient where both logs are in base 10. This video will show you how to apply the formula

log_ba = log_ca / log_cb

where a, b and c are all positive numbers and b, c are not equal to 1.

Word Problems and Logarithms (06:20) - This video will show you how to solve application problems involve logarithms and the decibal logs.

Domain of Log Functions (#) - This video will show you how to find the domain of a logarithmic function. Consider an example like "Find the domain of log (2x - 8). Since the argument of a log function must be greater than 0, we know 2x - 8 must be greater than 0. Solving for x, you get x > 4. That makes the domain all reals greater than 4. This video will show you how to find the domain of a logarithmic function.

Graphing (Linear, Quadratic Exponential and Logarithmic Functions)

Graphing Linear Functions (08:52) - This video talks about how to set up a table of values and graph straight lines of the form y = mx + b.

Graphing Quadratic Functions (17:29) - This video talks about finding the vertex of a parabola and using that to create the table of values necessary to graph a quadratic function.

Graphing Exponential Functions (15:46) - This video talks about graphing exponential functions.

Graphing Logarithmic Functions (11:37) - This video talks about graphing logarithmic functions and the importance of being able to find the domain!

Circles

Finding the Center and Radius of a Circle (03:16) - The formula for the equation of a circle in standard form, is (x-h)² + (y-k)² = r². The ordered pair (h,k) is the center of the circle and r is the radius. Be careful here. The minus signs built into the formula are separate from the values of h and k.

Finding the Equation of a Circle (05:55) - This video shows you how to find the equation of a circle in standard form by completing the squares. If you don't remember how to complete the squares, you may want to watch that video first.