Which of the following quartic functions has x = 2 and x = -3 as its only two real zeros?
a. x^4 - 5x^3 + 7x^2 - 5x + 6
b. x^4 + 5x^3 + 7x^2 + 5x + 6
c. x^4 + 5x^3 + 7x^2 - 5x - 6
d. x^4 - 5x^3 + 7x^2 + 5x - 6
To determine which quartic function has x = 2 and x = -3 as its only two real zeros, we need to test each option by plugging in these values and see if they equal to zero.
a. x^4 - 5x^3 + 7x^2 - 5x + 6:
When x = 2:
2^4 - 5(2)^3 + 7(2)^2 - 5(2) + 6 = 16 - 40 + 28 - 10 + 6 = 0
When x = -3:
(-3)^4 - 5(-3)^3 + 7(-3)^2 - 5(-3) + 6 = 81 + 135 + 63 + 15 + 6 = 300 ≠ 0
b. x^4 + 5x^3 + 7x^2 + 5x + 6:
When x = 2:
2^4 + 5(2)^3 + 7(2)^2 + 5(2) + 6 = 16 + 40 + 28 + 10 + 6 = 100 ≠ 0
When x = -3:
(-3)^4 + 5(-3)^3 + 7(-3)^2 + 5(-3) + 6 = 81 - 135 + 63 - 15 + 6 = 0
c. x^4 + 5x^3 + 7x^2 - 5x - 6:
When x = 2:
2^4 + 5(2)^3 + 7(2)^2 - 5(2) - 6 = 16 + 40 + 28 - 10 - 6 = 68 ≠ 0
When x = -3:
(-3)^4 + 5(-3)^3 + 7(-3)^2 - 5(-3) - 6 = 81 - 135 + 63 + 15 - 6 = 18 ≠ 0
d. x^4 - 5x^3 + 7x^2 + 5x - 6:
When x = 2:
2^4 - 5(2)^3 + 7(2)^2 + 5(2) - 6 = 16 - 40 + 28 + 10 - 6 = 8 ≠ 0
When x = -3:
(-3)^4 - 5(-3)^3 + 7(-3)^2 + 5(-3) - 6 = 81 + 135 + 63 - 15 - 6 = 258 ≠ 0
The quartic function that has x = 2 and x = -3 as its only two real zeros is b. x^4 + 5x^3 + 7x^2 + 5x + 6.