A polynomial with roots a and b is (x - a)(x - b).
(x - 2)(x - (-1))(x - 4) = 0
(x - 2)(x + 1)(x - 4) = 0
has roots 2, -1, and 4.
15 divided by 3 is 5
15 - 3=13
Factors of 24.
1, 2, 3, 4, 6, 8, 12, and 24<span>. </span>
You did it right (almost, I got 21 instead of 19) but didn't finish. You need to show your discriminant is never negative.
x² + (p+1)x = 5-2p
x² + (p+1)x +(2p-5) =0
Real roots mean a positive (or at least non-negative) discriminant:
D = b² - 4ac = (p+1)² - 4(1)(2p - 5) = p² + 2p + 1 - 8p + 20
D = p² - 6p + 21
It's not totally obvious that D>0; we prove that by completing the square by noting
(p-3)² = p² - 6p + 9
so
p² - 6p = (p-3)² - 9.
D = p² - 6p + 21
D = (p-3)² - 9 + 21
D = (p-3)² + 12
Now we clearly see D>0 always because the squared term can't be negative, so D is always at least 12. We always get two distinct real roots.
