Problem 72 Factor completely, or state that... [FREE SOLUTION]

Chapter 0: Problem 72

Factor completely, or state that the polynomial is prime. $$x^{3}+3 x^{2}-25 x-75$$

Short Answer

Expert verified

So, the complete factorization of the original polynomial $x^{3}+3 x^{2}-25 x-75$ is $(x-5)(x-3)(x-2)(x+2.5)$.

Step by step solution

Identify potential roots

The potential roots of a polynomial can be found as the factors of the constant term. In this case, the constant term is -75. So the factors are $\pm1, \pm3, \pm5, \pm15, \pm25, \pm75$.

Plug in potential roots

Each potential root can be plugged into the original polynomial, if it evaluates to zero, then it is a root. By calculating, we find that -5 and 3 are roots of the polynomial.

Polynomial division

Next, we perform synthetic division with the actual polynomial using the roots -5 and 3. When we perform the synthetic division with -5, we get another polynomial $x^{2}+x-15$, which when divided by 3 results in $x^{2}+x-5$.

Factorize the quotient polynomial

The final polynomial $x^{2}+x-5$ can be factored further by identifying factors that multiply to -5 and add to 1, which are 2 and -2.5. So it factorizes to become $(x-2)(x+2.5)$.

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91影视

Factor completely, or state that the polynomial is prime. $$x^{3}+3 x^{2}-25 x-75$$

Short Answer

Step by step solution

Identify potential roots

Plug in potential roots

Polynomial division

Factorize the quotient polynomial

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