The Zero-Product Property

The Zero-Product Property


– WELCOME TO A LESSON
ON THE ZERO PRODUCT PROPERTY. THE GOAL OF THIS VIDEO IS TO
USE THE ZERO PRODUCT PROPERTY TO SOLVE POLYNOMIAL EQUATIONS
IN FACTORING FORM. SO THE ZERO PRODUCT PROPERTY
IS ONE OF THE MAIN REASONS WHY WE LEARN HOW TO FACTOR
QUADRATIC EQUATIONS AND POLYNOMIAL EQUATIONS. THE ZERO PRODUCT PROPERTY
STATES, “IF TWO NUMBERS, A AND B
ARE MULTIPLIED TOGETHER “AND THE RESULTING PRODUCT
IS ZERO, THEN AT LEAST ONE
OF THE NUMBERS MUST BE ZERO.” SO IF A x B=0 THEN EITHER
A MUST=0 OR B MUST=0 OR BOTH A AND B ARE=TO 0. AND WE CAN USE THIS IDEA TO HELP US SOLVE POLYNOMIAL
EQUATIONS IN FACTORED FORM. IF WE WANT TO SOLVE
THE EQUATION X x THE QUANTITY X -3=0, BECAUSE THIS PRODUCT IS=TO 0 EITHER THE FIRST FACTOR OF X
MUST=0 OR THE SECOND FACTOR
OF X -3 MUST=0. SO WE KNOW ONE SOLUTION
IS X=0. AND THE SECOND SOLUTION WE HAVE TO SOLVE THIS EQUATION
FOR X SO WE’D ADD 3
TO BOTH SIDES OF THE EQUATION. SO – 3 + 3 IS=TO 0. SO OUR SECOND SOLUTION
IS X=+3. AGAIN, WE HAVE TWO SOLUTIONS. X=0 OR X=3. OF COURSE, IF WE WANTED TO
WE COULD CHECK THIS. TO CHECK X=0 WE WOULD
SUBSTITUTE 0 FOR X, WE WOULD HAVE
0 x THE QUANTITY 0 – 3. WELL, THAT WOULD BE 0 x -3
WHICH=0. THAT CHECKS. AND WHEN X IS=TO 3
WE’D SUBSTITUTE 3 FOR X, WE WOULD HAVE 3 x 3 – 3. WELL, 3 – 3 IS=TO 0. SO HERE WE WOULD HAVE 3 x 0,
WHICH ALSO=0 AND THEREFORE CHECKS. LET’S TAKE A LOOK
AT SOME MORE EXAMPLES. IN THIS EQUATION WE HAVE
4 X x THE QUANTITY X + 5=0. AGAIN, BECAUSE THIS PRODUCT
IS=TO 0 EITHER THE FIRST FACTOR
OF 4 X MUST=0 OR THE SECOND FACTOR
OF X + 5 MUST=0. AND NOW WE NEED TO SOLVE
EACH OF THESE EQUATIONS FOR X TO DETERMINE OUR SOLUTIONS. SO HERE TO ISOLATE X WE WOULD
DIVIDE BOTH SIDES BY 4. SO THIS WOULD BE 1 X OR JUST
X=0 DIVIDED BY 4 IS 0. TO SOLVE THIS EQUATION FOR X WE WOULD SUBTRACT 5 ON BOTH
SIDES +5 – 5 IS=TO 0. SO WE HAVE X=-5. AGAIN, WE HAVE 2 SOLUTIONS,
X=0 OR X=-5. HERE WE HAVE THE QUANTITY X –
2 x THE QUANTITY X + 7=0. AGAIN, BECAUSE THIS PRODUCT
IS=TO 0 EITHER X – 2 MUST=0
OR X + 7 MUST=0. AND NOW WE’LL SOLVE
THESE EQUATIONS FOR X. SO HERE WE ADD 2 TO BOTH SIDES
OF THE EQUATION, – 2 + 2 IS 0. SO WE’RE LEFT WITH X=+2
OR SOLVING THIS EQUATION FOR X WE WOULD SUBTRACT 7
ON BOTH SIDES OF THE EQUATION WHICH WOULD GIVE US X=-7. SO THESE ARE THE 2 SOLUTIONS
TO OUR POLYNOMIAL EQUATION IN FACTORED FORM. SO HOPEFULLY
NOW YOU’RE BEGINNING TO SEE WHY IT’S BENEFICIAL TO HAVE
A POLYNOMIAL EQUATION IN FACTORED FORM. LET’S TAKE A LOOK
AT TWO MORE EXAMPLES. HERE WE HAVE
THE QUANTITY 2 X + 3 x THE QUANTITY 5 X – 1=0. SO EITHER THE FIRST FACTOR
OF 2 X + 3 MUST=0 OR THE SECOND FACTOR
OF 5 X – 1 MUST=0. AND NOW WE’LL SOLVE
THESE EQUATIONS FOR X. SO HERE WE WOULD START
BY SUBTRACTING 3 ON BOTH SIDES OF THE EQUATION. THIS WOULD GIVE US 2 X=-3 AND THE LAST STEP HERE IS TO DIVIDE BOTH SIDES
OF THE EQUATION BY 2. THIS WOULD BE 1 X
OR X=-3 HALVES OR SOLVING THIS EQUATION FOR X WE WOULD START BY ADDING 1
TO BOTH SIDES OF THE EQUATION, THIS WOULD BE 0. SO IF 5 X EQUALS 1
AND DIVIDE BOTH SIDES BY 5. SO OUR SECOND SOLUTION
IS X=1/5th.   LET’S TAKE A LOOK
AT ONE MORE EXAMPLE. NOTICE IN THIS EQUATION
WE HAVE 3 FACTORS THAT HAVE A PRODUCT OF ZERO. SO IN THIS CASE WE’LL HAVE
3 SOLUTIONS EITHER X=0
FROM THIS FIRST FACTOR OR X – 1=0
FROM THE SECOND FACTOR OR 6 X + 11 IS=TO 0. AND NOW WE’LL SOLVE THESE
FOR X. WELL, THE FIRST EQUATION
IS ALREADY SOLVED FOR X. WE HAVE X=0. THE SECOND EQUATION,
WE’LL ADD 1 TO BOTH SIDES. THIS WILL GIVE US THE SOLUTION
X=+1. AND THEN
FOR THE THIRD EQUATION WE HAVE A 2 STEP EQUATION SO WE’LL SUBTRACT 11
ON BOTH SIDES AND THEN DIVIDE BOTH SIDES
BY 6. SO OUR THIRD SOLUTION
IS X=-11/6. SO AS LONG AS WE HAVE
OUR PRODUCT=TO 0 WE CAN TAKE ADVANTAGE
OF THE ZERO PRODUCT PROPERTY TO SOLVE
THE POLYNOMIAL EQUATION. SO BECAUSE OF THE ZERO PRODUCT
PROPERTY WE WILL SPEND SOME TIME
LEARNING HOW TO FACTOR A VARIETY
OF POLYNOMIALS SO THAT WE CAN SOLVE
POLYNOMIAL EQUATIONS. I HOPE YOU FOUND THIS HELPFUL.  

24 thoughts on “The Zero-Product Property

  1. There is one fallacy by multiplier zero:
    5 x 0 = 0  and  5 x 0 =  5   Both are correct because one factor represents an entity or attribute while the other factor is only a multiplier.
    There is a difference between an entity ( or attribute) and multiplier.
    Universally 5 x 0 = 0 because in computer logic gate AND does not differentiate between entity (or attribute) and multiplier of entity.

    Kindly  respond.

  2. my God. this helped me So Much! I missed the lesson and didn't know what the hell everyone was talking about. Thank you SO MUCH!

  3. Thanks, This makes a TON more sense..
    Its kinda funny that a video made in 2011 still helps people every week 😀

Leave a Reply

Your email address will not be published. Required fields are marked *