Factor Theorem

[Perji]

I have been presented with a problem which involves using the factor theorem. The problem states,

$$2x^2+x+1$$ is a factor of $$p(x)=2x^3-5x^2-2x+b$$.Find b and the roots of $$p(x)=0$$
Usually with these problems the factor is in the form $$(x-c)$$ I have tried to further factor and solve the given factor of $$p(x)$$ although it cannot be factored any further and the solutions give a complex number. Could anyone kindly give me some guidance involving this problem?
Thank you!

 

[Dr.Peterson]

I have been presented with a problem which involves using the factor theorem. The problem states,

$2x^2+x+1$ is a factor of $p(x)=2x^3-5x^2-2x+b$. Find b and the roots of $p(x)=0$

Usually with these problems the factor is in the form $(x-c)$ I have tried to further factor and solve the given factor of $p(x)$ although it cannot be factored any further and the solutions give a complex number. Could anyone kindly give me some guidance involving this problem?
Thank you!

If $(2x^2+x+1)q(x)=2x^3-5x^2-2x+b$, what does $q(x)$ look like? There are several facts you can easily see (starting with its degree). Once you see that, finding b will be very easy.

 

[JeffM]

It is difficult to know what answer will be meaningful to you without knowing what you know about the Fundamental Theorem of Algebra.

Let’s start here.

$$\text {If } p(x) = 2x^3 - 5x^2 - 2x + b,$$
what is the maximum number of distinct linear polynomials that can factor p(x)?

What is the minimum number of linear polynomials with a real root that can factor p(x)?

$$\text {If } r(x) = 2x^2 + x + 1 \text { is a factor of } p(x),$$
are the linear factors of r(x) also linear factors of p(x)?

Are there distinct linear factors of r(x)? What are they? Do they have real roots?

If not, how might you find q(x), a linear factor of p(x) with a real root (assuming such a beast exists)?