Mat 108 Intermediate AlgebraUnit 4 Part 6Higher Degree Equations in One VariableThe 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 VARIABLEEXAMPLES: (1) 3223242xxxxx(2) 481x(3) 323227525752xxxxxx(4) 421090xxRECOGNITION:(1) The variable appears with an exponent largerthan 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 replacementsfor xwill make the valueof the expression3232xxxequal to the value of the expression 242xxWHAT DOES IT LOOK LIKE: (Example 1) We first enter the expression 3232xxxas 1Yand then the expression 242xxas2Y. 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 algebraicallythat there are actually three solutions. 1
RULES:The rules are the same as they are for any equationOUTCOME: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 generallya predictor of the number of solutions. Therefore, the solution set usuallycontains the same number of solutions as the largestexponent in the equation.STRATEGY: Example 1:Solve: 3223242xxxxxAs with quadratic equation in one variable, the first step for solving this new type is to collect all terms on one side of the equal symbol leaving zeroon the other (in standard form).