Algebra Period 4

ZERO PRODUCTS PROPERTY: 6-8

A new friend that you can ALWAYS count on!

For any two rational numbers a and b, if ab = 0,
then either a = 0, b = 0 or both equal zero
HOW DOES OUR NEW FRIEND HELP US?
Try to solve x² + 10x + 24 = 0
You'll -24 from each side: x² + 10x  = -24
Now what????
It's a quadratic (x² term).
You can't isolate x because there's another term with x² in it.
We need the Zero Products Property to solve quadratics.


Using this property, we can solve quadratic equations by factoring the equation and setting each factor to zero and solving.

EXAMPLE: (5x + 1)(x - 7) = 0

Using the Zero Products Property, we know that either 5x - 1 must equal zero
or x - 7 must equal zero.
(That's the only way the product could be zero.)

Set both equal to zero and then solve:

if 5x + 1 = 0 then x must be -1/5

if x - 7 = 0 then x must be 7

If you substitute these answers for x back in the original equation, they will both end up as 0 = 0

(IN OTHER WORDS, THE SOLUTIONS WORK!)


In the first example, the equation was already factored.
Now you will first need to factor the equation (if possible), and then set each factor equal to zero to solve.
MAKE SURE YOU MOVE EVERYTHING TO ONE SIDE OF THE EQUATION!!!!!!
YOU MUST HAVE ZERO ON ONE SIDE OF THE EQUATION TO USE THE ZERO PRODUCTS PROPERTY!

EXAMPLE: x² = 16

First move the 16 to the left side: x²2 - 16 = 0

Now factor as the difference of two squares: (x + 4)(x - 4) = 0

Set each factor equal to zero and solve: x + 4 = 0 or x - 4 = 0

x = -4 or x = 4
Why do you have to have zero on one side? Just think about the simple equation
ab = 24.
Can you solve this? a could be 1 and then b would be 24. But a could be 3 and then b would be 8. Or a could be 4 and be would be 6. Or a could be 5/7 and then be would be 168/5. There are infinite possibilities!
The only way you can find 2 answers definitively is if the product is 0.
If ab = 0, then for sure either a = 0 or b = 0.
Otherwise the product could not be 0! 



NEW ALGEBRA TERM TO LEARN!

root: any solution that turns the equation into the value of zero is called a root of the polynomial, or a ZERO of the polynomial because when you're graphing the polynomial the y value is zero at this point

So if the directions say "Find the roots of......." it just means get zero on one side of the equation, factor, set each factor equal to zero, and solve.

SUMMARY OF SOLVING QUADRATICS: 
solutions = roots = zeros of the polynomial
1. Get zero on one side of the equation
2. Factor

3. Set each piece (factor) equal to zero

4. Solve as one or two step

5. Check by substituting in the ORIGINAL equation