Factor the polynomial P(x)=x^3-2x^2-13x-10. Then solve the equation P(x)=0.
This is what I got:
P(x)=(x+1)(x^2-3x-10)
P(x)=(x+1)(x-5)(x+2)
When it says solve the equation P(x)=0 does it means solve for x?
If so then x=-1 or 5 or -2 Correct???
please and thank you
you are correct.
Yes, you are correct!
To factor the polynomial P(x), we start by looking for its rational roots using the Rational Root Theorem. The possible rational roots of P(x) are factors of the constant term (-10) divided by factors of the leading coefficient (1). In this case, the factors of -10 are ±1, ±2, ±5, ±10, and the factors of 1 are ±1.
By testing these possible rational roots using synthetic division or long division, we can find that x = -1, 5, and -2 are indeed solutions of P(x) = 0.
So, the factored form of P(x) is P(x) = (x + 1)(x - 5)(x + 2).
To solve the equation P(x) = 0, you simply set each factor equal to zero:
x + 1 = 0 -> x = -1
x - 5 = 0 -> x = 5
x + 2 = 0 -> x = -2
Therefore, the solutions to the equation P(x) = 0 are x = -1, 5, and -2, as you correctly stated.