If two of the zeros of the cubic polynomial ax³ + bx² + cx + d are each equal to zero, find the third zero

Video Explanation

Watch the video explanation below:

Given

f(x) = ax³ + bx² + cx + d

Two of the zeros of the polynomial are:

0 and 0

To Find

The third zero of the polynomial.

Solution

Let the three zeros of the polynomial be:

0, 0 and γ

For a cubic polynomial ax³ + bx² + cx + d, the product of the zeros is:

−d / a

Step 1: Use the Formula for Product of Zeros

Product of the zeros = 0 × 0 × γ

= 0

According to the formula:

0 = −d / a

Step 2: Find the Value of d

−d / a = 0

⇒ d = 0

Step 3: Use the Factor Theorem

Since 0 is a zero of multiplicity 2,

x² is a factor of the polynomial.

So the polynomial can be written as:

f(x) = x²(ax + b)

Step 4: Find the Third Zero

ax + b = 0

∴ x = −b/a

Final Answer

The third zero of the polynomial is −b/a.

Conclusion

Hence, if two of the zeros of the cubic polynomial ax³ + bx² + cx + d are each equal to zero, then the third zero is −b/a.