How do you find all the zeros of f(x)= (x-1)(3x-4)(x^2+3)f(x)=(x1)(3x4)(x2+3)?

2 Answers
Jun 15, 2016

If the domain is real numbers, zeros are x=1x=1 and x=4/3x=43. If domain is extended to complex numbers, we have two additional zeros x=isqrt3x=i3 and x=-isqrt3x=i3.

Explanation:

Zeros of a polynomial are those values of variable, for which value of the polynomial becomes zero.

As the given polynomial f(x)=(x-1)(3x-4)(x^2+3_f(x)=(x1)(3x4)(x2+3 is already written as product of binomials, f(x)f(x) will be zero if

x-1=0x1=0 or 3x-4=03x4=0 or x^2+3=0x2+3=0.

If the domain for xx is real numbers, the two zeros are given by first two binomials i.e. x=1x=1 and x=4/3x=43 (as in such a case x^2+3x2+3 will always be non-zero.

However, if domain is extended to complex numbers, x^2+3=0x2+3=0 gives us two more zeros given by x=isqrt3x=i3 and x=-isqrt3x=i3.

Jun 15, 2016

The zeros, which correspond to the factors are:

x = 1x=1, x = 4/3x=43, x = +-sqrt(3)ix=±3i

Explanation:

f(x) = (x-1)(3x-4)(x^2+3)f(x)=(x1)(3x4)(x2+3)

Note that if any one of the factors (x-1)(x1), (3x-4)(3x4) or (x^2+3)(x2+3) is zero, then their product f(x)f(x) is zero.

If all of the factors are non-zero, then their product f(x)f(x) is also non-zero.

In symbols we could write:

f(x) = 0 <=> ((x-1) = 0 vv (3x-4) = 0 vv (x^2+3) = 0)f(x)=0((x1)=0(3x4)=0(x2+3)=0)

Looking at each factor in turn:

(x-1) = 0(x1)=0 if and only if x = 1x=1

(3x-4) = 0(3x4)=0 if and only if x = 4/3x=43

In general, note that each linear factor corresponds to a zero:

(x-a) = 0(xa)=0 if and only if x = ax=a

(qx-p) = 0(qxp)=0 if and only if x = p/qx=pq

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The last factor (x^2+3)(x2+3) is slightly more complicated.

First note that it has no Real zeros, since for any Real value of xx we have x^2 >= 0x20, so (x^2+3) >= 3(x2+3)3.

If we look at Complex numbers, we find:

(sqrt(3)i)^2 = (sqrt(3))^2 * i^2 = 3*(-1) = -3(3i)2=(3)2i2=3(1)=3

So:

x^2+3 = x^2-(-3) = x^2-(sqrt(3)i)^2 = (x-sqrt(3)i)(x+sqrt(3)i)x2+3=x2(3)=x2(3i)2=(x3i)(x+3i)

We now have linear factors, corresponding to zeros:

x = sqrt(3)i" "x=3i and " "x = -sqrt(3)i x=3i