Question

After factoring the following polynomial completely, use the Fundamental-Theorem of Algebra

Answer 1

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

The polynomial

The polynomial is given as:

$P(x) = (x + 7)(x - 2)\²(x^2 - 4)(x + 1)^2$

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

$P(x) = (x + 7)(x - 2)\²(x^2 - 2^2)(x + 1)^2$

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

So, we have:

$P(x) = (x + 7)(x - 2)\²(x - 2)(x + 2)(x + 1)^2$

Group the common factors

$P(x) = (x + 7)(x - 2)^3(x + 2)(x + 1)^2$

Multiplicity

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.

The number of roots

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

$n = 1 + 3 + 1 + 2$

$n = 7$

Hence, the polynomial has 7 roots

Read more about polynomials at:

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