Obtain all zeroes of the polynomial f(x) = 2x⁴ + x³ − 14x² − 19x − 6, if two of its zeroes are −2 and −1

Video Explanation

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Given

f(x) = 2x⁴ + x³ − 14x² − 19x − 6

Two zeroes of the polynomial are:

x = −2 and x = −1

To Find

All the zeroes of the given polynomial.

Solution

Step 1: Use Factor Theorem

Since −2 and −1 are zeroes of f(x),

(x + 2) and (x + 1) are factors of the polynomial.

Step 2: Divide f(x) by (x + 2)

Using synthetic division:

−2 | 2    1    −14    −19    −6
        −4    6    16    −32
————————————————
     2    −3    −8    −3    0

Quotient obtained:

2x³ − 3x² − 8x − 3

Step 3: Divide the Quotient by (x + 1)

Using synthetic division:

−1 | 2    −3    −8    −3
        −2    5    −5
———————————–
     2    −5    −3    0

New quotient:

2x² − 5x − 3

Step 4: Factorise the Quadratic Polynomial

2x² − 5x − 3 = 0

Splitting the middle term:

2x² − 6x + x − 3 = 0

2x(x − 3) + 1(x − 3) = 0

(2x + 1)(x − 3) = 0

∴ x = −1/2 or x = 3

Final Answer

All the zeroes of the given polynomial are:

−2, −1, −1/2 and 3

Conclusion

Hence, the polynomial f(x) = 2x⁴ + x³ − 14x² − 19x − 6 has four zeroes: −2, −1, −1/2 and 3.