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If all the zeros of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign. Is the statement true or false? Justify your answer

Solution:

Given, the zeros of a cubic polynomial are negative.

We have to determine whether all the coefficients and the constant term of the polynomial have the same sign.

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

Given, the roots are -𝛼, -ꞵ, -𝛾

Sum of the roots, 𝛼 + ꞵ + 𝛾 = -𝛼 - ꞵ - 𝛾 = -(𝛼 + ꞵ + 𝛾)

-(𝛼 + ꞵ + 𝛾) = -b/a

𝛼 + ꞵ + 𝛾 = b/a > 0

So, a and b have the same sign.

Sum of the product of two zeros at a time, 𝛼ꞵ + ꞵ𝛾 + 𝛾𝛼 = (-𝛼)(-ꞵ) + (-ꞵ)(-𝛾) + (-𝛾)(-𝛼)

𝛼ꞵ + ꞵ𝛾 + 𝛾𝛼 = c/a > 0

So, a and c have the same sign.

Product of all zeros, 𝛼ꞵ𝛾 = (-𝛼)(-ꞵ)(-𝛾)

-(𝛼ꞵ𝛾) = -d/a

𝛼ꞵ𝛾 = d/a > 0

So, a and d have the same sign.

Therefore, a, b, c and d all have the same signs

✦ Try This: If all the zeros of a cubic polynomial 8x³ + 5x² + 3x + 2 are negative, then all the coefficients and the constant term of the polynomial have the same sign. Is the statement true or false? Justify your answer

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

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

If all the zeros of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign. Is the statement true or false? Justify your answer

Summary:

If all the zeros of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign. The statement is true

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