Find all zeros of the polynomial x⁴ + x³ − 34x² − 4x + 120, if two of its zeros are 2 and −2

Video Explanation

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Given

f(x) = x⁴ + x³ − 34x² − 4x + 120

Two zeros of the polynomial are:

x = 2 and x = −2

To Find

All the zeros of the given polynomial.

Solution

Step 1: Use the Given Roots to Form a Factor

Since 2 and −2 are zeros of f(x),

(x − 2) and (x + 2) are factors of f(x).

Multiply them:

(x − 2)(x + 2) = x² − 4

Step 2: Divide f(x) by (x² − 4)

Divide x⁴ + x³ − 34x² − 4x + 120 by x² − 4:

First term:

x⁴ ÷ x² = x²

Multiply:

x²(x² − 4) = x⁴ − 4x²

Subtract:

[x⁴ + x³ − 34x²] − [x⁴ − 4x²] = x³ − 30x²

Bring down remaining terms:

x³ − 30x² − 4x + 120

Next term:

x³ ÷ x² = x

Multiply:

x(x² − 4) = x³ − 4x

Subtract:

[x³ − 30x² − 4x] − [x³ − 4x] = −30x²

Bring down +120:

−30x² + 120

Next term:

−30x² ÷ x² = −30

Multiply:

−30(x² − 4) = −30x² + 120

Subtract:

(−30x² + 120) − (−30x² + 120) = 0

So the remainder is 0. This means x² − 4 divides f(x) exactly.

Quotient obtained:

x² + x − 30

Step 3: Factorise the Quadratic Quotient

x² + x − 30 = 0

We look for two numbers whose product is −30 and sum is +1:

They are +6 and −5.

∴ x² + x − 30 = (x + 6)(x − 5)

Thus the remaining zeros are:

x = −6 and x = 5

Final Answer

All the zeroes of the given polynomial are:

2, −2, −6 and 5

Conclusion

Hence, the polynomial x⁴ + x³ − 34x² − 4x + 120 has four zeroes: 2, −2, −6 and 5.