Verify that 1/2, 1 and −2 are the zeros of the cubic polynomial f(x) = 2x³ + x² − 5x + 2 and verify the relationship between zeros and coefficients

Video Explanation

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Solution

Given polynomial:

f(x) = 2x³ + x² − 5x + 2

Step 1: Verify the Given Zeros

(i) For x = 1/2:

f(1/2) = 2(1/8) + 1/4 − 5(1/2) + 2

= 1/4 + 1/4 − 5/2 + 2

= 1/2 − 1/2 = 0

∴ 1/2 is a zero of f(x)

(ii) For x = 1:

f(1) = 2(1) + 1 − 5 + 2

= 0

∴ 1 is a zero of f(x)

(iii) For x = −2:

f(−2) = 2(−8) + 4 + 10 + 2

= −16 + 16

= 0

∴ −2 is a zero of f(x)

Hence, 1/2, 1 and −2 are the zeros of the given polynomial.

Step 2: Verify the Relationship Between Zeros and Coefficients

Let the zeros be α = 1/2, β = 1, γ = −2

Sum of zeros:

α + β + γ = 1/2 + 1 − 2 = −1/2

−b/a = −1/2

✔ Verified

Sum of products of zeros taken two at a time:

αβ + βγ + γα

= (1/2 × 1) + (1 × −2) + (−2 × 1/2)

= 1/2 − 2 − 1

= −5/2

c/a = −5/2

✔ Verified

Product of zeros:

αβγ = (1/2)(1)(−2) = −1

−d/a = −1

✔ Verified

Final Answer

The given numbers 1/2, 1 and −2 are the zeros of the cubic polynomial f(x) = 2x³ + x² − 5x + 2.

The relationship between zeros and coefficients is verified.

Conclusion

Thus, the given cubic polynomial satisfies all standard relationships between its zeros and coefficients.