Tip:
Highlight text to annotate it
X
- NOW, WE'LL LOOK AT SOME EXAMPLES OF FACTORING A SUM
IN DIFFERENCE OF CUBES.
NOTICE BOTH THE SUM AND DIFFERENCE OF CUBES
FACTORS INTO A BINOMIAL AND A TRINOMIAL.
SO IF WE HAVE A BINOMIAL IN THE FORM OF A CUBED + B CUBED,
NOTICE HOW THE FIRST BINOMIAL FACTOR IS A + B,
AND THE TRINOMIAL FACTOR IS A SQUARED - AB + B SQUARED.
AND IF WE HAVE A BINOMIAL IN THE FORM A CUBED - B CUBED,
THE FIRST BINOMIAL FACTOR IS A - B,
AND THE TRINOMIAL FACTOR IS A SQUARED + AB + B SQUARED.
SO IF WE WANT TO FACTOR X CUBED + 64,
WE NEED TO RECOGNIZE THAT WE HAVE A SUM OF PERFECT CUBES.
IT MIGHT BE HELPFUL TO WRITE THIS
IN THE FORM OF X RAISED TO THE THIRD POWER + 64's
A PERFECT CUBE BECAUSE IT'S EQUAL TO 4
RAISED TO THE THIRD POWER.
IN THIS FORM WE CAN RECOGNIZE
THAT A IS GOING TO BE EQUAL TO X AND B IS GOING TO BE EQUAL TO 4.
AND NOW WE CAN APPLY OUR FACTORING FORMULA.
AGAIN, THIS WILL FACTOR INTO A BINOMIAL TIMES A TRINOMIAL.
THE BINOMIAL FACTOR WILL BE A + B.
SO IN THIS CASE, WE'LL HAVE X + 4.
THE FIRST TERM OF THE TRINOMIAL FACTOR WILL BE "A" SQUARED.
SO IF A = TO X, A SQUARE WOULD BE X SQUARED - A x B.
WELL, IF A = X AND B = 4,
A x B WOULD BE X x 4 OR 4X + B SQUARED.
WELL, IF B = 4, B SQUARED = 16.
IF WE NEED TO, WE CAN MULTIPLY THIS OUT
TO VERIFY THAT IT IS EQUAL TO X CUBED + 64.
LOOKING AT THE SECOND EXAMPLE, WE HAVE 8X CUBED - 27.
SO THE FIRST THING TO RECOGNIZE HERE IS THAT
WE HAVE A DIFFERENCE OF PERFECT CUBES.
8X CUBED IS A PERFECT CUBE
BECAUSE WE CAN WRITE THIS AS 2X TO THE THIRD.
AND 27 IS A PERFECT CUBE
BECAUSE WE CAN WRITE THIS AS 3 TO THE THIRD.
SO IN THIS FORM WE KNOW THAT A MUST EQUAL 2X
AND B MUST EQUAL 3.
AND NOW WE CAN FACTOR OUR DIFFERENCE OF CUBES
INTO A BINOMIAL AND A TRINOMIAL.
THE BINOMIAL FACTOR, WHEN WE HAVE A DIFFERENCE OF CUBES
IS A - B.
SO HERE WE'LL HAVE 2X - 3.
THE FIRST TERM OF THE TRINOMIAL FACTOR WILL BE A SQUARED.
WELL, IF A = 2X, 2X SQUARE WOULD BE 4X SQUARED + A X B
WOULD BE 2X x 3 OR 6X + B SQUARED.
IF B = 3, B SQUARE WOULD BE 3 SQUARED OR 9.
OKAY. I HOPE YOU FOUND THIS HELPFUL.