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

Video Explanation

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Given

f(x) = 2x³ + x² − 5x + 2

The given zeroes are:

1/2, 1 and −2

To Verify

• That the given numbers are zeroes of f(x)
• The relationship between zeroes and coefficients

Solution

Step 1: Verify the Zeroes

(i) f(1/2)

f(1/2) = 2(1/2)³ + (1/2)² − 5(1/2) + 2

= 2(1/8) + 1/4 − 5/2 + 2

= 1/4 + 1/4 − 5/2 + 2

= 1/2 − 5/2 + 2 = 0

∴ 1/2 is a zero of f(x).

(ii) f(1)

f(1) = 2(1)³ + (1)² − 5(1) + 2

= 2 + 1 − 5 + 2 = 0

∴ 1 is a zero of f(x).

(iii) f(−2)

f(−2) = 2(−2)³ + (−2)² − 5(−2) + 2

= −16 + 4 + 10 + 2 = 0

∴ −2 is a zero of f(x).

Hence, the given numbers are verified as zeroes of the polynomial.

Step 2: Verify Relationship Between Zeroes and Coefficients

For a cubic polynomial ax³ + bx² + cx + d:

Sum of zeroes = −b/a

Sum of product of zeroes taken two at a time = c/a

Product of zeroes = −d/a

Here, a = 2, b = 1, c = −5, d = 2

(i) Sum of Zeroes

1/2 + 1 + (−2) = −1/2

−b/a = −1/2

✔ Verified

(ii) Sum of Products of Zeroes Taken Two at a Time

(1/2 × 1) + (1 × −2) + (−2 × 1/2)

= 1/2 − 2 − 1 = −5/2

c/a = −5/2

✔ Verified

(iii) Product of Zeroes

(1/2) × 1 × (−2) = −1

−d/a = −2/2 = −1

✔ Verified

Final Answer

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

The relationship between zeroes and coefficients is verified.

Conclusion

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