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A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Give an example and explain why a polynomial can have fewer x-intercepts than its number of roots.
Solution:
Let us consider a fourth degree polynomial
f(x) = x4 - x3 - x2 - x - 2
From the remainder theorem,
f(-1) = 1 + 1 - 1 + 1 - 2 = 0
So (x + 1) is a factor.
f(2) = 16 - 8 - 4 - 2 - 2 = 0
So (x - 2) is a factor.
(x + 1)(x - 2) = x2 - 2x + x - 2
= x2 - x - 2
Use long division method to find the remaining factors
f(x) = (x + 1)(x - 2)(x2 + 1)
We know that (x2 + 1) contains no real factors
x2 + 1 = (x + i)(x - i)
It has a pair of conjugate zeros +i and - i.
Therefore, f(x) = x4 - x3 - x2 - x - 2 is an example and has fewer x-coordinates.
Give an example and explain why a polynomial can have fewer x-intercepts than its number of roots.
Summary:
f(x) = x4 - x3 - x2 - x - 2 is an example and has fewer x-intercepts than its number of roots.
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