IA In-Class 4-6 - Mat 108 Intermediate Algebra Unit 4 Part...

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Mat 108 Intermediate Algebra Unit 4 Part 6 Higher Degree Equations in One Variable The last topic for this unit is that of higher degree equations in one variable. By our definition, these are equations where the exponent with the variable is greater than 2 . HIGHER DEGREE EQUATIONS IN ONE VARIABLE EXAMPLES: (1) 3 2 2 3 2 4 2 x x x x x + + + = - + (2) 4 81 x = (3) 3 2 3 2 2 7 5 25 7 5 2 x x x x x x + - - = + - + (4) 4 2 10 9 0 x x - + = RECOGNITION: (1) The variable appears with an exponent larger than two (2) There are two expressions linked by an equal sign (3) There is only one variable (it may appear several times) WHAT DOES IT ASK : (Example1) What replacements for x will make the value of the expression 3 2 3 2 x x x + + + equal to the value of the expression 2 4 2 x x - + WHAT DOES IT LOOK LIKE: (Example 1) We first enter the expression 3 2 3 2 x x x + + + as 1 Y and then the expression 2 4 2 x x - + as 2 Y . After hitting the GRAPH key, we see an “S-curve” and a “U-shaped” curve that appear to intersect at one point as seen below (the TI is in Zoom 6 – the standard window): Since the two curves appear to intersect at only one point, we might think that there is only one solution to this equation. We will find out later when we solve the equation algebraically that there are actually three solutions. 1
RULES: The rules are the same as they are for any equation OUTCOME: The German mathematician Gauss proved the Fundamental Theorem of Algebra about two hundred years ago and one consequence of his theorem is that the highest exponent in an equation (with one variable) is generally a predictor of the number of solutions. Therefore, the solution set usually contains the same number of solutions as the largest exponent in the equation.

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