Zero Factor Property
Key Questions
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You use the zero factor property after you have factored the quadratic to find the solutions.
It is best to look at an example:
#x^2+x-6=0# This factors into:
#(x+3)(x-2)=0# We find our solutions by setting each factor to zero and solve:
#x+3=0#
#x=-3# or
#x-2=0#
#x=2# Previous answer (I was thinking some more complicated before):
You are not using the words precisely. You use the factor theorem with the factor property. The factor theorem states that if you find a
#k# such that#P(k)=0# , then#x-k# is a factor of the polynomial. The factor property states that#k# must a factor of the constant term in#P(x)# .Having said all that, you wouldn't normally use the factor theorem or factor property to solve a quadratic; they are many used to find factors of higher order polynomials. Once you reduce the higher order polynomial to a quadratic, you use regular factoring methods such as FPS or PFS: Factors, Product, and Sum.
#P(x)=ax^2+bx+c# The problem with the factor theorem and factor property is that it's not as easy to use when
#a!=1# .
Amory W.
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Aug 18 2014
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Answer:
Well, if you a polynomial is factorable then its roots/zeroes can be easily found by setting it to zero and using the zero factor property. Please see explanation below.
Explanation:
The Zero Product Property:
A product of factors is zero if and only if one or more of the factors is zero. Or:
if#a*b = 0# , then either#a = 0# or#b = 0# or both.
Example: Find the roots of the polynomial by factoring:
#P(x) = x^3-x^2-x+1# , set to zero:
#x^3-x^2-x+1=0# , factor by grouping:
#x^2(x-1)-1(x-1)=0#
#(x^2-1)(x-1)=0# , use difference of squares to factor further:
#(x+1)(x-1)(x-1)=0# , use the zero factor property:
#x+1=0=>x=-1#
#x-1=0=>x=1#
Notice that#x = 1# has a multiplicity of 2.
iceman
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2
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Sep 24 2015
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The zero factor property states that if
#ab=0# , then either#a=0# or#b=0# .Example: find the roots of
#x^2-x-6# .#x^2-x-6=0# #(x-3)(x+2)=0# Now, the zero factor property can be applied, since two thing are being multiplied and equal zero.
We know that either
#x-3=0color(white)(ssss)# or#color(white)(ssss)x+2=0# Solve both to find that
#x=3# or#x=-2# .
mason m
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1
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Jan 9 2016