Question

solve each equation given then indicated root
x^3 + 5x^2 -29x - 105 = 0, x=3

Answer 1

I get to 1x^2 + 2x + 35 but then I dont know what to do

Answer 2

The roots of a polynomial are those values for which the polynomial = 0. Furthermore, you get one root (which may be repeated) for each power of the variable in the polynomial. For example, your original polynomial is \[x^3+5x^2-29x-105\]which will have 3 roots because the highest power of \(x\) appearing in the polynomial is 3. We can also write the polynomial as a product of terms:

\[P(x) = x^3+5x^2-29x-105 = (x-r_1)(x-r_2)(x-r_3)\] where \((r_1,r_2,r_3)\) are the roots. A little bit of thought should show that wherever x = one of the roots, that product will equal 0, and the product will equal 0 only where x = one of the roots.

One of the implications of this is that if we know the value of a root (such as we do in this case), we can divide \(P(x)\) by \((x-r_n)\) to simplify the search for the remaining roots.

[by the way, it looks like you may have copied the problem slightly incorrectly: the indicated root should be \(x=-3\), I think]

Answer 3

Assuming the root is really \(x=-3\), we divide \[\frac{P(x) }{ (x-(-3))} = \frac{x^3+5x^2-29x-105}{(x+3)} = x^2+2x-35\]

(check by multiplying:)
\[(x+3)(x^2+2x-35) = x^3+2x^2-35x+3x^2+6x-105 = x^3+5x^2-29x-105\]

Now we need to find the roots of our simplified polynomial, \[P_1(x) = x^2+2x-35\]

We could do that by factoring, completing the square, or using the quadratic formula.