Which of the following statements are true andwhich are false ? Give a short proof or acounter-example, whichever is appropriate insupport of your answer : 20(i) A cubic equation with real coefficients hasat least one real root
Question
Which of the following statements are true andwhich are false ? Give a short proof or acounter-example, whichever is appropriate insupport of your answer : 20(i) A cubic equation with real coefficients hasat least one real root
Solution
To determine whether the statement is true or false, we need to provide a short proof or a counter-example.
The statement (i) "A cubic equation with real coefficients has at least one real root" is true.
Proof: Let's consider a general cubic equation with real coefficients in the form of ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are real numbers.
To find the roots of this equation, we can use the fact that any polynomial equation with real coefficients will have complex roots in conjugate pairs. Therefore, if the equation has no real roots, it must have a pair of complex conjugate roots.
Now, let's assume that the cubic equation has no real roots. This means that all the roots are complex conjugate pairs. Let's say the complex roots are p + qi and p - qi, where p and q are real numbers and i is the imaginary unit.
By Vieta's formulas, the sum of the roots of a cubic equation is given by -b/a and the product of the roots is given by -d/a.
Since the equation has no real roots, the sum of the roots will be 0 (since the sum of complex conjugate pairs is always 0). Therefore, -b/a = 0, which implies b = 0.
However, this contradicts the assumption that the equation has real coefficients, as the coefficient b cannot be zero. Hence, our assumption that the cubic equation has no real roots must be false.
Therefore, we can conclude that a cubic equation with real coefficients must have at least one real root.
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