If x^4+ax^3+bx^2+cx+d=0 has four positive real roots then
A) ac+16d>=0
B) ac-16d<=0
C) b^2-36d<=0
D) b^2+36d>=0
Urgent plse asap
If you expand (x-m)(x-n)(x-p)(x-q) you get a set of positive k values
x^4 - k1 x^3 + k2 x^2 - k3 x + k4
That means that ac > 0 and of course, d > 0
So, A is looking good.
A little experimentation will show that the others are not always true.
To determine the relationship between the coefficients of the equation and the roots, we can use Vieta's formulas. Vieta's formulas state that for an equation of the form:
ax^4 + bx^3 + cx^2 + dx + e = 0,
the sum of the roots is equal to -b/a, the sum of the products of the roots taken two at a time is equal to c/a, and so on.
Since the equation in question has four positive real roots, we know that the sum of the roots is positive. Let's denote the roots as r1, r2, r3, and r4.
According to Vieta's formulas, we have:
r1 + r2 + r3 + r4 = -a
Since the roots are positive, we can conclude that -a < 0, which means a > 0.
Now, let's consider the product of the roots taken two at a time:
r1r2 + r1r3 + r1r4 + r2r3 + r2r4 + r3r4 = c
Since all the roots are positive, each term in the above equation is positive. Therefore, we can deduce that c > 0.
Based on these conclusions, we can eliminate options A) and C) since ac + 16d >= 0 and b^2 - 36d <= 0, respectively.
To further deduce the relationship between the remaining options, let's consider another property of the given equation. If we substitute x = 0 into the equation, the constant term d is equal to 0. Therefore, we can conclude that d = 0.
Now, let's calculate the discriminant of the quadratic equation formed by taking the square of the sum of the two roots and comparing it to four times the product of the roots:
(r1r2 + r1r3 + r1r4 + r2r3 + r2r4 + r3r4)^2 - 4(r1r2r3 + r1r2r4 + r1r3r4 + r2r3r4) = b^2
Since all the roots are positive, all the terms in the above equation are positive. This implies that b^2 > 0, which means b^2 + 36d > 0.
Therefore, the correct answer is option D) b^2 + 36d >= 0.
To solve this question, we need to examine the relationship between the coefficients of the equation and the roots. In particular, we'll look at Vieta's formulas, which relate the coefficients of a polynomial equation to the sums and products of its roots.
Let's go through the steps:
Step 1: Vieta's Formulas
According to Vieta's formulas, the sum of the roots of a polynomial equation of the form axⁿ + bxⁿ⁻¹ + cxⁿ⁻² + ... + qx + r = 0 is given by the expression: (-1)ⁿ⁻¹ * (ⁿ-₁c₁/c₀).
For the given equation x⁴ + ax³ + bx² + cx + d = 0, the sum of the roots would be -a.
Step 2: Relation between Coefficients and Roots
Now, we'll use the information given in the question: the equation has four positive real roots. This implies that each root will contribute positively to the sum.
Since the sum of the roots is -a, and all the roots are positive, we have -a > 0, which means a < 0.
Step 3: Examining the Options
Now, let's evaluate each option based on the relationship between the coefficients:
A) ac + 16d ≥ 0
We know that a < 0, so the first term will be negative. There is no particular constraint on the sign of 16d. Therefore, ac + 16d may or may not be greater than or equal to 0. This option does not necessarily hold true.
B) ac - 16d ≤ 0
Again, a < 0, so the first term will be negative. There is no particular constraint on the sign of 16d. Therefore, ac - 16d may or may not be less than or equal to 0. This option does not necessarily hold true.
C) b² - 36d ≤ 0
There is no constraint given on the value of b. However, since we have four positive real roots, the discriminant of the quadratic equation formed by the roots (b² - 4ac) should be greater than or equal to 0. Since it is a quartic equation, the discriminant can be expressed as b² - 4ad, which simplifies to b² - 36d. Therefore, this option holds true.
D) b² + 36d ≥ 0
Similar to option C, the discriminant should be greater than or equal to 0. However, this option requires b² + 36d to be greater than or equal to 0, which does not necessarily hold for a quartic equation. This option does not necessarily hold true.
Step 4: Conclusion
Based on our analysis, the correct option is:
C) b² - 36d ≤ 0, as it is the only option that is guaranteed to be true for a quartic equation with four positive real roots.
Please note that this explanation assumes the given information is correct, and all roots are distinct.