Obtain all zeroes of the polynomial f(x) = x³ + 13x² + 32x + 20, if one of its zeroes is −2

Video Explanation

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Given

f(x) = x³ + 13x² + 32x + 20

One zero of the polynomial is:

x = −2

To Find

All the zeroes of the given polynomial.

Solution

Step 1: Use Factor Theorem

Since −2 is a zero of f(x),

(x + 2) is a factor of the polynomial.

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

Using synthetic division:

−2 | 1    13    32    20
        −2    −22    −20
———————————–
     1    11    10    0

Quotient obtained:

x² + 11x + 10

Step 3: Factorise the Quadratic Polynomial

x² + 11x + 10 = 0

Splitting the middle term:

x² + 10x + x + 10 = 0

x(x + 10) + 1(x + 10) = 0

(x + 10)(x + 1) = 0

∴ x = −10 or x = −1

Final Answer

All the zeroes of the given polynomial are:

−2, −10 and −1

Conclusion

Hence, the polynomial f(x) = x³ + 13x² + 32x + 20 has three zeroes: −2, −10 and −1.