If 4 is a zero of the cubic polynomial x³ − 3x² − 10x + 24, find its other two zeroes

Video Explanation

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Solution

Given polynomial:

f(x) = x³ − 3x² − 10x + 24

Given that x = 4 is a zero of the polynomial.

Step 1: Verify x = 4 Using Factor Theorem

f(4) = (4)³ − 3(4)² − 10(4) + 24

= 64 − 48 − 40 + 24

= 0

∴ x − 4 is a factor of f(x)

Step 2: Divide the Polynomial by (x − 4)

x³ − 3x² − 10x + 24 ÷ (x − 4)

Using synthetic division:

4 | 1   −3   −10   24
        4   4   −24
——————————–
     1   1   −6   0

∴ Quotient = x² + x − 6

Step 3: Find the Remaining Zeroes

Solve:

x² + x − 6 = 0

Factorising:

(x + 3)(x − 2) = 0

∴ x = −3 or x = 2

Final Answer

The other two zeroes of the given polynomial are:

x = −3 and x = 2

Conclusion

Thus, the zeroes of the cubic polynomial x³ − 3x² − 10x + 24 are 4, −3 and 2.