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- IN THIS VIDEO, I'D LIKE TO REVIEW
MULTIPLYING BINOMIALS WITH YOU.
A BINOMIAL IS SIMPLY A POLYNOMIAL
THAT HAS TWO PIECES OR TWO TERMS.
SO, FOR EXAMPLE: (X + 4)(X - 2).
SO WHAT WE WOULD LIKE TO DO IS WE'D LIKE TO
MULTIPLY THESE TOGETHER.
HOW DO WE KNOW WE'RE MULTIPLYING THEM?
BECAUSE WHEN WE LOOK IN-BETWEEN THESE PARENTHESES,
WE DON'T SEE ANY SYMBOL.
WE DON'T SEE A PLUS SIGN, SO WE'RE NOT ADDING.
WE DON'T SEE A MINUS SIGN, SO WE'RE NOT SUBTRACTING.
WHEN THEY'RE RIGHT NEXT TO EACH OTHER,
THAT MEANS WE'RE MULTIPLYING.
NOW, ESSENTIALLY, WHEN YOU MULTIPLY TWO BINOMIALS TOGETHER,
YOU NEED TO DISTRIBUTE TWICE.
OFTEN WHAT I FIND IS STUDENTS WHO ARE TAUGHT THIS
IN HIGH SCHOOL HAVE BEEN TAUGHT AN ACRONYM
OR AN ABBREVIATION CALLED FOIL.
AND SO I'M GOING TO DO IT THAT WAY WITH YOU
BECAUSE THAT MIGHT BE FAMILIAR.
FOIL STANDS FOR FIRST, OUTSIDE, INSIDE AND LAST.
LET ME WRITE IT OVER HERE WHERE MORE OF YOU CAN MAYBE SEE IT.
SO FOIL.
SO FIRST, OUTSIDE, INSIDE AND LAST.
AND IT'S SIMPLY A WAY TO REMIND YOU
THAT YOU REALLY HAVE TO DISTRIBUTE TWICE,
AND IF YOU GO THROUGH LETTER BY LETTER,
THEN YOU WON'T FORGET ANY PART OF THE DISTRIBUTING YOU NEED.
I WANT TO DO ONE THAT'S A LITTLE HARDER WITH YOU,
ONE WHERE WE MIGHT HAVE A NUMBER HERE IN FRONT OF THE X,
JUST TO MAKE SURE THAT THAT WON'T HANG YOU UP IN ANY WAY.
SO WHAT IF WE HAD SOMETHING LIKE (2X - 3)(3X + 2)?
AGAIN, IF YOU WANT TO REMEMBER THE ACRONYM FOIL
OR THE ABBREVIATION FOIL, WE'RE GOING TO GO FOR THE FIRST.
2 x 3 = 6, X, X-- IN OTHER WORDS, 6X SQUARED.
THE OUTSIDE ONES, 2X AND 2.
THAT'S GOING TO BE A +4X.
THE INSIDE ONES, A -3 AND A 3. THAT'S -9 AND AN X.
AND FINALLY THE LAST ONES.
SO -3 x 2 = -6.
AND THEN NOTICE THAT, AGAIN, THESE TWO HERE,
WE CAN COMBINE TOGETHER.
SO WE'RE GOING TO HAVE 6X SQUARED.
4X - 9X = -5X, AND HERE WE HAVE A -6.
SO OUR ANSWER WOULD BE 6X SQUARED - 5X - 6.
SO WE DID AN EASY ONE.
WE DID ONE THAT'S A LITTLE HARDER.
I WANT TO REMIND YOU
THAT THERE'S NO SHORTCUTS ON THESE.
SOMETIMES WHEN YOU'RE GIVEN A MULTIPLICATION
OF BINOMIAL PROBLEM THAT'S NOT WRITTEN OUT,
PEOPLE ARE TEMPTED TO TAKE A SHORTCUT.
THAT WILL ALWAYS GIVE YOU THE WRONG ANSWER.
SO WHAT IF WE HAD (X + 3) SQUARED?
PEOPLE OFTEN MAKE THE MISTAKE OF THINKING
THAT YOU CAN JUST SQUARE X AND SQUARE 3
AND YOU GET X SQUARED + 9.
THAT SHORTCUT DOESN'T WORK. NO SHORTCUTS.
SO IF YOU SHOULD SEE A PROBLEM LIKE THIS,
MAKE SURE TO WRITE IT OUT.
THIS MEANS WRITE THIS OUT TWICE: (X + 3)(X + 3).
AND THEN ONCE THAT'S DONE, YOU'RE GOING TO, AGAIN,
USE YOUR FIRST, OUTSIDE, INSIDE, LAST.
SO THE FIRST ONES: X x X = X SQUARED.
THE OUTSIDE ONES: X x 3 = 3X.
THE INSIDE ONES: 3 x X = 3X.
AND THEN THE LAST ONES: 3 x 3 = 9.
AND WHAT WE SHOULD DO IS COMBINE THE SIMILAR TERMS IN HERE.
SO A FINAL ANSWER WOULD BE X SQUARED + 6X + 9.
SO HOPEFULLY YOU CAN SEE THAT IF YOU WERE
TO ACCIDENTALLY TAKE THE TEMPTING BUT WRONG SHORTCUT
OF JUST SQUARING X AND SQUARING THREE,
YOU WOULD BE MISSING THIS MIDDLE TERM HERE.
SO WHEN YOU SEE ONE LIKE THIS, I REALLY RECOMMEND
JUST WRITE IT OUT
AND THEN APPLY FOIL LIKE YOU ALWAYS WOULD.
ALL RIGHT. SO LET'S TRY THIS ONE.
WE'LL TRY AN EASY ONE, ONE THAT'S A LITTLE HARDER,
AND THEN ONE WHERE I WILL TEMPT YOU TO TRY A SHORTCUT,
BUT YOU'LL REMEMBER JUST TO WRITE IT OUT AND WORK IT OUT.
SO SEE IF YOU CAN DO THESE.
THERE'S AN EASY ONE.
LET'S DO ONE THAT'S A LITTLE HARDER,
AND THEN LET'S TRY THIS ONE.
ALL RIGHT. SO I HOPE YOU PRESSED PAUSE AND TRIED THEM.
LET'S GO AHEAD AND WORK THEM OUT.
SO ON THIS ONE, WE'LL MULTIPLY THE FIRST ONES TOGETHER.
X x X = X SQUARED.
WE'LL MULTIPLY THE OUTSIDE ONES TOGETHER,
AND THAT WILL GIVE US A -4X.
WE'LL MULTIPLY THE INSIDE ONES TOGETHER.
THAT WILL GIVE US A -6X.
AND FINALLY WE'LL MULTIPLY THESE LAST ONES TOGETHER.
NOW, DON'T FORGET,
IT'S A NEGATIVE TERM TIMES A NEGATIVE TERM,
SO THAT'S GOING TO GIVE US A +24.
AND IN THE END, CHECK FOR SIMILAR TERMS,
USUALLY RIGHT HERE IN THE MIDDLE.
WE HAVE X SQUARED - 10X + 24.
AND SO THAT'S YOUR ANSWER FOR THE FIRST ONE.
LET'S DO THE SECOND ONE THE SAME WAY.
WE'LL--FIRST ONES TOGETHER.
SO WE HAVE 3 x 4 = 12, AND THEN X, X--
IN OTHER WORDS, X SQUARED.
WE'LL MULTIPLY THE OUTSIDE ONES TOGETHER.
3 x -5 = -15, SO WE HAVE -15X.
WE'LL MULTIPLY THE INSIDE ONES TOGETHER.
1 x 4 = 4, SO WE HAVE 4X.
AND FINALLY WE'LL MULTIPLY THE LAST ONES TOGETHER.
1 x -5 = -5.
AGAIN, WE HAVE SIMILAR TERMS RIGHT HERE IN THE MIDDLE.
OUR FINAL ANSWER IS GOING TO BE 12X SQUARED - 11X - 5.
AND FINALLY, LET'S TAKE A LOOK AT THIS LAST ONE HERE.
IT WAS X + 6 TO THE SECOND POWER.
AND SO, REMEMBER, IT WILL BE TEMPTING
JUST TO SQUARE EACH OF THESE, BUT THAT'LL BE WRONG EVERY TIME.
IF WE WERE TO TAKE A MOMENT TO WRITE IT OUT, (X + 6)(X + 6),
AND WE WERE TO START WITH X x X, THAT'LL BE X SQUARED.
THE OUTSIDE ONES HERE WILL GIVE US A 6X.
THE INSIDE ONES HERE WILL GIVE US ANOTHER 6X.
AND 6 x 6 IS GOING TO BE 36.
COMBINING THE SIMILAR TERMS IN THE MIDDLE,
THE FINAL ANSWER WILL BE X SQUARED + 12X + 36.
AND THAT'S A QUICK RECAP OF MULTIPLYING BINOMIALS.