If all three zeros of a cubic polynomial x³ + ax² - bx + c are positive, then at least one of a, b and c is non-negative. Is the statement true or false? Justify your answer

Solution:

Given, the cubic polynomial is x³ + ax² - bx + c.

The zeros of a cubic polynomial are positive.

We have to determine at least a, b and c are non-negative.

We know that, if 𝛼, ꞵ and 𝛾 are the zeroes of a cubic polynomial ax³ + bx² + cx + d, then

𝛼 + ꞵ + 𝛾 = -b/a

𝛼ꞵ + ꞵ𝛾 + 𝛾𝛼 = c/a

𝛼ꞵ𝛾 = -d/a

Here, a = 1, b = a, c = -b and d = c

Sum of roots, 𝛼 + ꞵ + 𝛾 = -b/a = -a/1

𝛼 + ꞵ + 𝛾 = -a

a < 0

Sum of the product of two zeros at a time, 𝛼ꞵ + ꞵ𝛾 + 𝛾𝛼 = -b/1

𝛼ꞵ + ꞵ𝛾 + 𝛾𝛼 = -b > 0

b < 0

Product of all zeros, 𝛼ꞵ𝛾 = -c/1

𝛼ꞵ𝛾 = -c > 0

c < 0

Therefore, the cubic polynomial has all three zeros positive when all the constants a, b and c are negative.

✦ Try This: If all three zeros of a cubic polynomial x³ + sx² - tx + u are positive, then at least one of s, t and u is non-negative. Is the statement true or false? Justify your answer

Given, the cubic polynomial is x³ + sx² - tx + u.

The zeros of a cubic polynomial are positive.

We have to determine at least s, t and u are non-negative.

We know that, if 𝛼, ꞵ and 𝛾 are the zeroes of a cubic polynomial ax³ + bx² + cx + d, then

𝛼 + ꞵ + 𝛾 = -b/a

𝛼ꞵ + ꞵ𝛾 + 𝛾𝛼 = c/a

𝛼ꞵ𝛾 = -d/a

Here, a = 1, b = s, c = -t and d = u

Sum of roots, 𝛼 + ꞵ + 𝛾 = -b/a = -s/1

𝛼 + ꞵ + 𝛾 = -s

s < 0

Sum of the product of two zeros at a time, 𝛼ꞵ + ꞵ𝛾 + 𝛾𝛼 = -t/1

𝛼ꞵ + ꞵ𝛾 + 𝛾𝛼 = -t > 0

t < 0

Product of all zeros, 𝛼ꞵ𝛾 = -u/1

𝛼ꞵ𝛾 = -u > 0

u < 0

If all three zeros of a cubic polynomial x³ + sx² - tx + u are positive, then at least one of s, t and u is non-negative.

Therefore, the statement is false

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 2

NCERT Exemplar Class 10 Maths Exercise 2.2 Problem 2 (vi)

If all three zeros of a cubic polynomial x³ + ax² - bx + c are positive, then at least one of a, b and c is non-negative. Is the statement true or false? Justify your answer

Summary:

If all three zeros of a cubic polynomial x³ + ax² - bx + c are positive, then at least one of a, b and c is non-negative. The statement is false

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