unit 6 lesson 9 algebra 2 connexus
find the rational roots of x^4+8x^3+7x^2-40x-60=0
the Rational Root Theorem can narrow the choices.
A little synthetic division yields
(x+2)(x+6)(x^2-5) = 0
-6,-2
The rational roots of the equation x^4 + 8x^3 + 7x^2 - 40x - 60 = 0 are -6 and -2.
To find the rational roots of the given polynomial equation, x^4+8x^3+7x^2-40x-60=0, we can use the Rational Root Theorem. This theorem states that if a rational number p/q is a root of the polynomial equation with integer coefficients, then p must be a factor of the constant term of the polynomial (in this case, -60) and q must be a factor of the leading coefficient (in this case, 1).
Step 1: Find the factors of the constant term:
The constant term is -60, so we need to find all the factors of -60. The factors of -60 are:
±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60
Step 2: Find the factors of the leading coefficient:
The leading coefficient is 1, so the only possible rational roots are ±1.
Step 3: Test the possible roots:
Using synthetic division or long division, test each possible root by substituting it into the polynomial equation to check if it equals zero. If it does, then that value is a rational root.
Let's test the possible rational roots one by one:
If we substitute x = 1 into the polynomial, we get:
(1)^4 + 8(1)^3 + 7(1)^2 - 40(1) - 60 = 1 + 8 + 7 - 40 - 60 = -84
Since -84 is not zero, x = 1 is not a rational root.
If we substitute x = -1 into the polynomial, we get:
(-1)^4 + 8(-1)^3 + 7(-1)^2 - 40(-1) - 60 = 1 - 8 + 7 + 40 - 60 = -20
Since -20 is not zero, x = -1 is not a rational root.
None of the possible rational roots (±1 and the factors of -60) are actual roots of the polynomial equation. Therefore, there are no rational roots for the given equation x^4+8x^3+7x^2-40x-60=0.