7.1 Determine When Rational Expression is Not Defined

IN CHAPTER 7, WE ARE LOOKING AT RATIONAL EXPRESSIONS AND THAT'S REFERRING TO EXPRESSIONS THAT LOOK ALMOST LIKE A FRACTION. IN FACT, IT IS A FRACTION, BUT THE NUMERATOR AND DENOMINATOR ARE MADE UP OF POLYNOMIALS. SO THERE'S A FEW THINGS YOU NEED TO LEARN ABOUT RATIONAL EXPRESSIONS, SO LET'S BREAK IT DOWN THIS WAY. FIRST THING I NEED YOU TO KNOW IS HOW TO DETERMINE WHEN A RATIONAL EXPRESSION IS NOT DEFINED. SO, BEFORE I SHOW YOU HOW TO DO IT, LET'S RECALL SOMETHING YOU LEARNED IN ARITHMETIC ABOUT FRACTIONS. IF YOU HAVE 0 OVER 10, THAT'S EQUAL TO 0, BUT WHEN YOU HAVE 5 DIVIDED BY 0, FOR EXAMPLE, THAT'S GONNA BE UNDEFINED. SIMILARLY, IF YOU HAVE 0 OVER 0, THAT'S ALSO UNDEFINED. SO WHAT MAKES THESE TWO UNDEFINED IS BECAUSE THE DENOMINATORS IN BOTH CASES ARE EQUAL TO ZERO, SO HENCE WE SAY THE DENOMINATOR CANNOT BE ZERO BECAUSE, WHEN THAT HAPPENS, YOUR QUOTIENT IS UNDEFINED. OKAY, SO, IF THIS WAS SOMETHING WE WROTE BEFORE, BUT LET ME WRITE THIS AGAIN. A QUOTIENT OF TWO NUMBERS IS UNDEFINED WHENEVER THE DIVISOR IS ZERO. SO WHEN THE DENOMINATOR IS A ZERO, REGARDLESS WHAT THE NUMERATOR IS, YOUR QUOTIENT WILL BE UNDEFINED. SO HOW WILL WE ASK YOU THE QUESTION IN CHAPTER 7? THE WAY WE'LL ASK YOU IS, CAN YOU DETERMINE THE VALUES THAT MAKE THE EXPRESSION 3X OVER 2 DIVIDED BY X SQUARED - 2X - 8 NOT DEFINED? SO IN SO MANY WORDS, WE'RE ASKING YOU WHAT VALUES MUST YOU PUT FOR X SO THAT--PUT INTO X--SO THAT YOUR QUOTIENT BECOMES UNDEFINED? SO RATHER THAN YOU SITTING THERE GUESSING WHAT NUMBER WORKS OR WHAT NUMBER MAKES THE DENOMINATOR ZERO, THE METHOD IS TO SET THE DENOMINATOR TO ZERO. SO YOU FORCE IT TO BE ZERO, SET DENOMINATOR EQUALS ZERO AND SOLVE IT. SO THAT'S THE METHOD. SO WE FORCE IT TO BE ZERO. SO IN OUR EXAMPLE, I'M GONNA SET X SQUARED - 2X - 8 = 0. SO THIS IS A TRINOMIAL, BACK TO WHAT YOU LEARNED IN CHAPTER 6. FOR TRINOMIAL, WE USE SUM AND PRODUCT. SUM IS -2, THAT'S THE MIDDLE, AND PRODUCT IS -8, SO YOU NEED TO ASK YOURSELF WHAT TIMES WHAT GIVES YOU 8, BUT YET THE SUM IS -2? SO THE ANSWER IS 4 AND 2 AND IT'S -4, +2, IF YOU WANT A SUM OF -2. SO FILL IN THE FACTORS X - 4, X + 2, NOT BECAUSE THE PRODUCT OF THIS IS ZERO, SO THAT'S WHEN YOU FORCE EACH PIECE TO ZERO. SO X - 4 = 0 TELLS YOU X IS 4 AND X + 2 = 0 TELLS YOU X IS -2. SO THE ANSWER TO THIS QUESTION IS 4, SO THE ANSWER TO THIS QUESTION DETERMINE THE VALUES IS 4, -2. OKAY. THAT'S THE ANSWER. BUT THE BIG QUESTION IS WHY? WHAT DOES THAT MEAN? OKAY, WHY? NOW WHEN X TAKES THE VALUE OF 4, YOU WILL NOTICE THE EXPRESSION 3X - 2 OVER X SQUARED - 2X - 8. THAT'S THE EXPRESSION WE ARE TALKING ABOUT RIGHT HERE. WHAT HAPPENS IS, YOU'RE GONNA REPLACE X BY 4, SO 3 x 4 - 2 OVER 4 SQUARED - 2 x 4 - 8. SO ON THE TOP IS GONNA BE A 12 - 2 WHICH IS A 10, THAT'S THE NUMERATOR, BUT IN THE DENOMINATOR IS A 16 - 8 - 8, WHICH ACTUALLY TURNS OUT TO BE 0. SO TRUE ENOUGH, WHEN X IS 4, THE VALUE IS UNDEFINED. OKAY, THE QUOTIENT IS UNDEFINED. SO WE ARE SAYING THAT WHAT VALUES MAKES IT UNDEFINED? 4 DOES, AND ON TOP OF THAT, -2 DOES AS WELL. OKAY, SO LET'S CHECK. 3 x -2 - 2 OVER -2 SQUARED - 2 x -2 - 8. SO 3 x -2 IS -6, -6 - 2, THAT MAKES THE TOP -8, BUT THE BOTTOM IS GOING TO BE A 4 + 4 - 8, WHICH IS A 0. SO TRUE ENOUGH, IT'S UNDEFINED AGAIN. IN ANSWER TO THIS QUESTION, WHAT ARE THE VALUES THAT MAKES IT UNDEFINED, WE ARE SAYING THAT THERE'S TWO NUMBERS THAT WILL MAKE IT UNDEFINED. THE TWO NUMBERS ARE 4 AND -2 AND YOU CAN SEE FROM HERE, IF YOU SUBSTITUTE 4 OR -2 INTO THE QUOTIENT, INTO WHERE X IS, THE QUOTIENT BECOMES UNDEFINED IN BOTH CASES. OKAY. SO THIS IS JUST SERVE AS A CHECK JUST SO WE UNDERSTAND WHAT WE ARE DOING, BUT THE METHOD IS REALLY TO LOOK FOR THE VALUES THAT MAKE THE DENOMINATOR EQUAL ZERO. SO WE FORCED IT TO MAKE IT EQUAL ZERO. THAT'S WHAT WE ARE DOING RIGHT HERE, THE METHOD, AND THEN WE SOLVE IT. SO LET'S LOOK AT ANOTHER EXAMPLE, SAME THING. CAN YOU DETERMINE THE VALUES THAT MAKE, LET'S TRY, 2X + 1 OVER -48 + 12X + 6X SQUARED NOT DEFINED? OKAY, SO IGNORING THE NUMERATOR. ALL WE DO IS SET THE DENOMINATOR WHICH IN THIS CASE IS -48 + 12X + 6X SQUARED AND WE SET IT EQUAL TO ZERO. SO IF YOU RECALL FOR TRINOMIALS, WE HAVE TO ARRANGE THEM IN DESCENDING ORDER OF EXPONENTS, SO WE START WITH 6X SQUARED FOLLOWED BY 12X - 48 = 0. THEN REMEMBER GCF. SO THE FIRST THING IS TO REARRANGE, SO I'M HOPING THAT THIS WILL DO A QUICK REVIEW FOR YOU. FIRST THING IS TO REARRANGE, THEN THE NEXT THING IS TO LOOK FOR GCF, IN THIS CASE, IT'S A 6. SO IT'S X SQUARED + 2X - 8 AND THEN, LOOKING AT THE TRINOMIAL, I'M LOOKING FOR A SUM OF +2 THIS TIME AND THE PRODUCT I'M LOOKING FOR IS -8. SO WE'RE STILL TALKING ABOUT 4 AND 2 EXCEPT THAT I NEED +4 THIS TIME AND -2 FOR THE SUM TO BE +2. SO WHEN I PUT IN THE FACTORS, IT'S GOING TO BE X + 4 AND X - 2. SO THIS IS WHERE YOU SET EACH PIECE TO ZERO, 6 = 0, X + 4 = 0, AND X - 2 = 0. IF YOU LOOK AT THE FIRST CASE, 6 = 0, THAT'S BASICALLY NONSENSE, SO THERE'S NO CONTRIBUTION FROM THERE, SO YOU REJECT IT. SO IT'S NONSENSE, SO WE SUBTRACT IT. X + 4 = 0 TELLS YOU X IS -4 AND X - 2 = 0 TELLS YOU X IS 2, SO THE ANSWER TO THIS QUESTION IS -4, 2. THOSE ARE THE TWO THAT WILL WORK. SO IF YOU LIKE, YOU CAN ALWAYS CHECK TO SEE THAT YOU ARE CORRECT. SO LET'S TRY ONE MORE. LET'S TRY A DIFFERENT WAY OF PHRASING. THIS IS ANOTHER WAY WE CAN SAY CAN YOU FIND THE VALUES OF THE VARIABLE FOR WHICH, LET'S TRY, 5X + 3 OVER 1 - 4X SQUARED IS NOT DEFINED? OKAY, SO THAT'S ANOTHER WAY WE CAN PHRASE THE QUESTION, BUT THE KEY IS WE ARE LOOKING FOR THE VALUES THAT MAKES IT NOT DEFINED. SO THE METHOD IS THE SAME. YOU'RE SUPPOSED TO SET THE DENOMINATOR WHICH, IN THIS CASE, 1 - 4X SQUARED. SET IT TO ZERO, EXCEPT THIS TIME IT'S A BINOMIAL. SO FOR A BINOMIAL, YOU HAVE TO THINK OF WHETHER IT BELONGS TO THE SQUARED GROUP OR THE CUBES. SO IN THIS CASE, IT BELONGS TO THE SQUARES, SO START TWO PARENTHESES. WHAT TIMES WHAT GIVES YOU A 1? THAT'S 1 x 1. 4X SQUARED IS 2X AND 2X AND THEN ONE PLUS AND ONE MINUS. OKAY, SO SAME THING. YOU WILL SET EACH PIECE TO ZERO. 1 + 2X = 0 TELLS YOU 2X IS -1 AND SO X IS -1 HALF. AND IN THIS CASE, 1 - 2X = 0 WHICH MEANS -2X IS -1, SO X IS -1 OVER -2 WHICH IS POSITIVE 1/2. SO THE ANSWER TO THIS QUESTION IS PLUS OR MINUS 1/2. SO THOSE ARE THE TWO NUMBERS THAT WILL MAKE IT NOT DEFINED. OKAY, YOUR TURN TO TRY. IF YOU LOOK INTO YOUR BOOK ON PAGE 300, YOU SHOULD WORK ON THESE NUMBERS, LET'S SEE, NUMBER 15, NUMBER 27 AND NUMBER 31, OKAY? ALL RIGHT. �