Question

Let t represent an unknown number. What is the value

Answer 1

There are two possible answers:

t = 2/3   or   t = -1/3

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Explanation:

The expression x+3 is the same as x-(-3). It is of the form x-k

Recall that by the remainder theorem, we know that p(x) is divisible by (x-k) if and only if p(k) = 0.

In this case, k = -3. Plug this into the p(x) function and we get

p(x) = t^2*x^3 - 3t*x + 6

p(-3) = t^2*(-3)^3 - 3t*(-3) + 6

p(-3) = -27t^2 + 9t + 6

Set that equal to 0

p(-3) = 0

-27t^2 + 9t + 6 = 0

-3(9t^2 - 3t - 2) = 0

9t^2 - 3t - 2 = 0

Now apply the quadratic formula to solve for t

$t = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\t = \frac{-(-3)\pm\sqrt{(-3)^2-4(9)(-2)}}{2(9)}\\\\t = \frac{3\pm\sqrt{81}}{18}\\\\t = \frac{3\pm9}{18}\\\\t = \frac{3+9}{18}\ \text{ or } \ t = \frac{3-9}{18}\\\\t = \frac{12}{18}\ \text{ or } \ t = \frac{-6}{18}\\\\t = \frac{2}{3}\ \text{ or } \ t = -\frac{1}{3}$

We get two possible solutions for t.