The image below demonstrates the factoring a fourth degree polynomial with four terms. The first step is to find the greatest common factor (GCF). Factor trees are used here to accomplish this. This technique comes in handy if you cannot come up with the GCF mentally. In the second step, a new binomial appears with its own GCF. We again use factor trees. In the third step, a difference of squares emerges. In the fourth step, the polynomial is completely factored. The next two images demonstrate the simplification of rational expressions. This time, factor trees are used slightly differently, to find the least common multiple (LCM) rather than the GCD (as used for factoring a polynomial expression). Each factor is circled in the tree where it is most numerous. All of the circled numbers are multiplied to yield the LCM, which is the lowest common denominator in the rational expression.
In the last step, we check to see if the trinomial in the numerator is factorable, which turns out it is not.
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If we start with the mixed number three and fiveeighths, it can be read as shorthand notation for the sum of three and five/eighths. This represents long division as follows. The whole number 3 is the quotient, the numerator 5 is the remainder, and the denominator 8 is the divisor. The mixed number can be converted to an improper fraction by multiplying quotient and divisor, adding the remainder, then dividing the quantity by the divisor. This can be represented vertically as a fraction, or horizontally as a standard order of operations. The result is 29/8, or 29 divided by 8. If long division is carried out next, the answer can be expressed in three different ways. If expressed as a decimal, the decimal point can be moved two places to the right, implicitly multiplying by 100, to get the equivalent percentage. Hence, the quantity can be expressed equivalently as a mixed number, improper fraction, decimal, or percentage. The next image shows various ways for the number to be expressed using pie and bar diagrams. Three and fiveeighths can be viewed as three whole pies or rectangles, with a fourth pie or rectangle divided into eight pieces with three pieces that are empty. As the above photo shows, the four rectangles can be stacked end to end. The pies or rectangles can be equivalently divided into 16 congruent pieces, with the fourth unit having six empty pieces. Hence, three and tensixteenths is equivalent to three and fiveeighths. The final image shows two of the whole units being stuck together, then converted to a sum of three numbers: two plus one plus fiveeighths. Using the associative property of addition, or the order of operations, the next step shows three plus fiveeights. In the final step, we return to the mixed number that we started with in this demonstration.

Mr. Ken Marciel
