Find all zeroes of the polynomial 3x³ + 10x² − 9x − 4, if one of its zeroes is 1

Video Explanation

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Solution

Given polynomial:

f(x) = 3x³ + 10x² − 9x − 4

Given that x = 1 is a zero of the polynomial.

Step 1: Verify x = 1 Using Factor Theorem

f(1) = 3(1)³ + 10(1)² − 9(1) − 4

= 3 + 10 − 9 − 4

= 0

∴ x − 1 is a factor of f(x)

Step 2: Divide the Polynomial by (x − 1)

3x³ + 10x² − 9x − 4 ÷ (x − 1)

Using synthetic division:

1 | 3   10   −9   −4
        3   13   4
——————————–
     3   13   4   0

∴ Quotient = 3x² + 13x + 4

Step 3: Find the Remaining Zeroes

Solve:

3x² + 13x + 4 = 0

Factorising:

3x² + 12x + x + 4 = 0

3x(x + 4) + 1(x + 4) = 0

(3x + 1)(x + 4) = 0

∴ x = −1/3 or x = −4

Final Answer

The zeroes of the given polynomial are:

x = 1, −1/3 and −4

Conclusion

Thus, all the zeroes of the polynomial 3x³ + 10x² − 9x − 4 are 1, −1/3 and −4.