9x^2-64 Factor

[Endike Baur]

Primenumbers are natural numbers (positive whole numbers that sometimes include 0 in certain definitions) that are greater than 1, that cannot be formed by multiplying two smaller numbers. An example of a prime number is 7, since it can only be formed by multiplying the numbers 1 and 7. Other examples include 2, 3, 5, 11, etc.

Prime numbers are widely used in number theory due to the fundamental theorem of arithmetic. This theorem states that natural numbers greater than 1 are either prime, or can be factored as a product of prime numbers. As an example, the number 60 can be factored into a product of prime numbers as follows:

One method for finding the prime factors of a composite number is trial division. Trial division is one of the more basic algorithms, though it is highly tedious. It involves testing each integer by dividing the composite number in question by the integer, and determining if, and how many times, the integer can divide the number evenly. As a simple example, below is the prime factorization of 820 using trial division:

Another common way to conduct prime factorization is referred to as prime decomposition, and can involve the use of a factor tree. Creating a factor tree involves breaking up the composite number into factors of the composite number, until all of the numbers are prime. In the example below, the prime factors are found by dividing 820 by a prime factor, 2, then continuing to divide the result until all factors are prime. The example below demonstrates two ways that a factor tree can be created using the number 820:

While these methods work for smaller numbers (and there are many other algorithms), there is no known algorithm for much larger numbers, and it can take a long period of time for even machines to compute the prime factorizations of larger numbers; in 2009, scientists concluded a project using hundreds of machines to factor the 232-digit number, RSA-768, and it took two years.

Why study quadratics? The graphs of quadratic equations result in parabolas (U shaped graphs that open up or down). This feature of quadratics makes them good models for describing the path of an object in the air or describing the profit of a company (examples of which you may see in Finite Mathematics or in Microeconomics.)

Example 1. A boy lying on his back uses a sling shot to fire a rock straight up in the air with an initial velocity (the force the boy uses to fire the rock) of 64 feet per second. The quadratic equation that models the height of the rock is

According to the graph, the rock reaches its greatest height at 2 seconds. The maximum height is 64 feet. The maximum or minimum point of a quadratic is called the vertex. You will learn how to find the vertex in Section 4.3, Quadratic Applications and Graphs.

According to the graph, the rock is on the ground at zero seconds (right before the boy shoots it) and at 4 seconds (when the rock lands). These points are the time intercepts. You will learn how to find them in the next Section 4.2, "Applications of the Quadratic Formula."

Couldn't combine the unlike terms inside the parantheses so we used the distributive property. After that, we multiplied 6x by 3 and then -5 by 3.

Used the distributive property and combined like terms.

Substituted the revenue and cost equations into the formula for profit. Must use parantheses.

Used the distributive property and multiplied the revenue equation by 1 and cost equation by -1.

Combined like terms.

Substituted the revenue and cost equations into the formula for profit. Must use parentheses. Used the distributive property.Multiplied the revenue equation by 1 and the cost equation by -1. Combined like terms.

Quadratics are important equations in physics and microeconomics. The technique for adding and subtracting quadratics is the same as we have been practicing all semester; that is, add or subtract the like terms. To multiply, use the distributive property or FOIL. The vertex of the quadratic will be explained in more detail in the section, "Graphing Quadratics and Applications." The vertex is the maximum or minimum point on the graph of the quadratic.

Example 1. Suppose you are standing on top of a cliff 375 feet above the canyon floor, and you throw a rock up in the air with an initial velocity of 82 feet per second. The equation that models the height of the rock above the canyon floor is:

Example 2. A rancher has 500 yards of fencing to enclose two adjacent pig pens that rest against the barn. If the area of the two pens must total 20,700 square yards, what should the dimensions of the pens be?

This section shows us how to solve a new type of equation, the quadratic. These have important applications in many fields, such as business, physics, and engineering. Learnthe difference between the quadratic equation and the quadratic formula.

Example 1. A boy lying on his back uses a sling shot to fire a rock straight up in the air with an initial velocity (the force the boy uses to shoot the rock) of 64 feet per second. The quadratic equation that models the height of the rock is

Explanation: One explanation for the profit having two break even points is how efficient a company is at making a product. Making very few items is usually inefficient. At some point, the factory becomes very efficient at manufacturing the product, but if the factory tries to make too many items, the company becomes inefficient at producing its product.

The company will earn a profit of more than $500,000 when the profit graph is above the horizontal line P = 500. This problem is similar to example 2d on page 203 in Section 2.9 "Applications of Graphs".

Explanation: The most difficult part of the table is finding the value for length. If the farmer uses 10 meters for the width of the pens, and there are 4 widths, then he has used 4 times 10 , or 40 meters of fencing. To find how much fencing he has left for the length, subtract 40 from 96, the total amount of fencing available to the farmer.

where T is measured in Celsius, and m represents the minutes that the experiment has run. Graph the equation by finding the vertex and the intercepts. Label these points on the graph and explain what the vertex and intercepts mean in terms of the model.

Study Tips: Quadratics are U shaped graphs. In some cases, they are U shaped as in the example above or shaped as in examples 1 through 3. If a in the equation, y = ax2 + bx + c, is positive, then the graph is U shaped, that is, opening up. If a is negative, the graph is shaped, that is, opening down. This fact should be written on a note card.

Factoring is an algebraic technique used to separate an expression into its component parts. When the component parts are multiplied together, the result is the original expression. This can sometimes be used to solve quadratic equations. Factoring is an important skill in MAT 100, Intermediate Algebra.

Before you think that factoring to solve quadratics is a lot easier than using the quadratic formula, you need to know that factoring doesn't always work. Consider changing Example 8 by just one to x2 - 11x + 31 = 0. You cannot find two integers that when added equal -11 and when multiplied equal 31. To factor x2 - 11x + 31 you must use the quadratic formula. You will learn how to factor any quadratic equation in Precalculus I, MAT 161.

Two techniques for factoring are presented in this unit. The first is common factors which uses the distributive property, ab + ac = a(b + c). The other one is factoring trinomials. To factor trinomials, you need to know how FOIL works. If you take MAT 100, Intermediate Algebra, you will see more factoring.

This chapter introduced you to quadratics. The two major topics are the quadratic formula and graphs of quadratics. These topics have many applications in business, physics, and geometry. Factoring is an important topic in MAT 100, Intermediate Algebra.

This is a factoring calculator if specifically for the factorization of the difference of two squares. If the input equation can be put in the form of a2 - b2 it will be factored. The work for the solution will be shown for factoring out any greatest common factors then calculating a difference of 2 squares using the idenity:

If a is negative and we have addition such that we have -a2 + b2 the equation can be rearranged to the form of b2 - a2which is the correct equation only the letters a and b are switched; we can just rename our terms.

Just enter any positive integer, and in the blink of an eye, you'll find all positive factors of that number. If you are not sure what a factor is, scroll down to find the factor definition, as well as divisibility rules with a dedicated paragraph for not so well-known divisibility rule of 7.

A factor, also called a divisor, is any number that divides evenly into another number. In other words, factors are the numbers we can multiply together to get a certain product:

Definition of the factor differs: some definitions claim that factor can be negative as well as positive, but in other cases, the term is restricted to positive factors only.

For example, the factors of 8 are 1, 2, 4, and 8. But, on the other hand, if you multiply -2 times -4, you'll also obtain 8; therefore, -2 and -4 are factors of 8 according to the first definition.

Technically, you can have negative factors, although it's not so popular to use them. For practical purposes, our factor calculator provides only positive factors. If you need negative ones for some reason, just add the minus in front of every obtained value:

Prime factorization is an extension of factorization in which all the factors are prime numbers. For example, suppose we want the prime factorization of 48. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Notice those are not all prime numbers, so we have to break it down further. When completing the process, we get 2 2 2 2 3. Although 1 is a factor, many mathematicians now do not consider 1 to be a prime number. The prime factorization calculator is a handy tool for obtaining these factors.

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