anemone said:Let $p,\,q,\,r,\,s,\,t$ be any real numbers and $s\ne 0$.
Prove that the equation $x^3+(p+q+r)x^2+(pq+qr+rp-s^2)x+t=0$ has at least two distinct roots.
kaliprasad said:let $P(x) = x^3 +(p+q+r)x^2 + (pq+qr+rp-s^2)x + t$
so $\dfrac{dP(x)}{dx} = 3x^2 + 2 (p+q+r) x + (pq+qr+rp-s^2)$
now discriminant
= $4(p+q+r)^2 - 12(pq+qr+rp-s^2)$
= $4(p^2+q^2+r^2 -pq - qr -rp + 3s^2)$
= $2((p-q)^2 + (q-r)^2+(r-p)^2 + 6s^2)$
now as s is not zero the discriminant is not zero or derivative does not have double root so P(x) cannot have 3 same roots hence it has at least 2 distinct roots
A cubic polynomial is a mathematical expression of the form ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and x is a variable. It is called cubic because the highest power of the variable is 3.
To solve a cubic polynomial, you can use the formula known as the cubic formula, which uses the coefficients a, b, c, and d to find the roots of the equation. You can also use other methods like factoring, graphing, or using the rational root theorem.
Distinct roots are solutions to a polynomial equation that are different from each other. In the case of a cubic polynomial, there can be up to 3 distinct roots.
To prove that a cubic polynomial has two distinct roots, you can use the discriminant b^2-4ac to determine the number of real roots. If the discriminant is positive, then there are two distinct real roots. You can also use other methods like factoring or graphing to show that there are two distinct roots.
Proving that a cubic polynomial has two distinct roots is important because it can help verify the accuracy of the solution and provide a better understanding of the behavior of the polynomial. It can also help in finding the exact values of the roots, which can be useful in various applications in science and engineering.