After factoring the following polynomial completely, use the Fundamental-Theorem of Algebra
to determine the number of roots of the polynomial. Then, determine which of the factors has a
multiplicity of 3 and compare it to the Code Breaker Guide to find the fifth piece of the code.
(x + 7)(x - 2)²(x2 – 4)(x + 1)^*

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MrRoyal MrRoyal · Answer 1 · 2022-01-11T10:23:43+01:00

The factor that has a multiplicity of 3 is x - 2, and the polynomial has 7 roots.

The polynomial is given as:

[tex]P(x) = (x + 7)(x - 2)\²(x^2 - 4)(x + 1)^2[/tex]

Express 4 as 2^2. So, the polynomial becomes

[tex]P(x) = (x + 7)(x - 2)\²(x^2 - 2^2)(x + 1)^2[/tex]

Apply the difference of two squares on (x^2 - 2^2).

So, we have:

[tex]P(x) = (x + 7)(x - 2)\²(x - 2)(x + 2)(x + 1)^2[/tex]

Group the common factors

[tex]P(x) = (x + 7)(x - 2)^3(x + 2)(x + 1)^2[/tex]

For a factor to have a multiplicity of 3, it means that the exponent (i.e. power) is 3.

This means that x - 2 has a multiplicity of 3.

Next, we add the multiplicities of each factor, to get the number of roots.

[tex]n = 1 + 3 + 1 + 2[/tex]

[tex]n = 7[/tex]

Hence, the polynomial has 7 roots