Zero-Product Property
New york state common core math algebra i, module 4, lesson 5.
Lesson 5 Summary
Zero Product Property
If ab = 0, then a = 0 or b= 0 or a = b = 0 When solving for the variable in a quadratic equation, rewrite the equation as a factored quadratic set equal to zero. Using the zero product property, you know that if one factor is equal to zero, then the product of all factors is equal to zero.
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Common Core Algebra 1
These interactive lessons use dynamic graphing and guided discovery to strengthen and connect symbolic and visual reasoning. They give the student a hands-on visual exposition of all Common Core Algebra 1 topics, reinforced by adaptive exercises and randomly generated tests. All exercises and tests are checked and graded automatically. Hover the mouse over a link below to see an example from that lesson, or click on a test link to see a concise summary of a group of lessons. Relevant standards are listed after each lesson’s exercises, with a ' (prime) denoting a distinct link. We also provide a course glossary .
The Common Core and other standards cover many algebraic topics before the first actual High School Algebra course. Therefore, students may have already completed many of the lessons and exercises below in an earlier grade, or in a summer preparatory course . If so, those assignments will be treated as already completed in this course also.
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Variables, expressions, and simple equations, linear graphs, inequalities, absolute value, and square roots, descriptive statistics, systems of linear equations or inequalities, quadratic expressions, graphs, and equations, the pythagorean theorem, exponential growth and decay.
Zero Product Principle
Anything multiplied by zero is zero.
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Zero Product Property
The "Zero Product Property" says that:
If a × b = 0 then a = 0 or b = 0 (or both a=0 and b=0)
It can help us solve equations:
Example: Solve (x−5)(x−3) = 0
The "Zero Product Property" says:
If (x−5)(x−3) = 0 then (x−5) = 0 or (x−3) = 0
Now we just solve each of those:
For (x−5) = 0 we get x = 5
For (x−3) = 0 we get x = 3
And the solutions are:
x = 5 , or x = 3
Here it is on a graph:
Standard Form of an Equation
Sometimes we can solve an equation by putting it into Standard Form and then using the Zero Product Property:
The "Standard Form" of an equation is:
(some expression) = 0
In other words, "= 0" is on the right, and everything else is on the left.
Example: Put x 2 = 7 into Standard Form
x 2 − 7 = 0
Standard Form and the Zero Product Property
So let's try it out:
Example: Solve 5(x+3) = 5x(x+3)
It is tempting to divide by (x+3), but that is dividing by zero when x = −3
So instead we can use "Standard Form":
5(x+3) − 5x(x+3) = 0
Which can be simplified to:
(5−5x)(x+3) = 0
5(1−x)(x+3) = 0
Then the "Zero Product Property" says:
(1−x) = 0, or (x+3) = 0
x = 1 , or x = −3
And another example:
Example: Solve x 3 = 25x
It is tempting to divide by x, but that is dividing by zero when x = 0
So let's use Standard Form and the Zero Product Property.
Bring all to the left hand side:
x 3 − 25x = 0
Factor out x:
x(x 2 − 25) = 0
x 2 − 25 is a difference of squares , and can be factored into (x − 5)(x + 5) :
x(x − 5)(x + 5) = 0
Now we can see three possible ways it could end up as zero:
x = 0 , or x = 5 , or x = −5
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Common Core Algebra II.Unit 6.Lesson 1.Quadratic Function Review
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Oct 20, 2016
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This unit begins with a fundamental treatment of exponent rules and the development of negative and zero exponents. We then develop the concepts of exponential growth and decay from a fraction perspective. Finally, percent work allows us to develop growth models based on constant percent rates of change. Geometric sequences are tied to exponential growth in the last lesson.
Simplifying Expressions Involving Exponents
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