Verify that 2, 1 and 1 are the zeros of the cubic polynomial g(x) = x³ − 4x² + 5x − 2 and verify the relationship between zeros and coefficients

Video Explanation

Watch the video explanation below:

Solution

Given polynomial:

g(x) = x³ − 4x² + 5x − 2

Step 1: Verify the Given Zeros

(i) For x = 2:

g(2) = (2)³ − 4(2)² + 5(2) − 2

= 8 − 16 + 10 − 2

= 0

∴ 2 is a zero of g(x)

(ii) For x = 1:

g(1) = 1 − 4 + 5 − 2

= 0

∴ 1 is a zero of g(x)

(iii) For x = 1 (repeated root):

g(1) = 0

∴ 1 is a repeated zero of g(x)

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

Step 2: Verify the Relationship Between Zeros and Coefficients

Let the zeros be α = 2, β = 1 and γ = 1

Sum of zeros:

α + β + γ = 2 + 1 + 1 = 4

−b/a = −(−4)/1 = 4

✔ Verified

Sum of products of zeros taken two at a time:

αβ + βγ + γα

= (2 × 1) + (1 × 1) + (1 × 2)

= 2 + 1 + 2

= 5

c/a = 5/1 = 5

✔ Verified

Product of zeros:

αβγ = 2 × 1 × 1 = 2

−d/a = −(−2)/1 = 2

✔ Verified

Final Answer

The given numbers 2, 1 and 1 are the zeros of the cubic polynomial g(x) = x³ − 4x² + 5x − 2.

The relationship between zeros and coefficients is verified.

Conclusion

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