Given that √2 is a zero of the cubic polynomial 6x³ + √2x² − 10x − 4√2, find its other two zeroes

Video Explanation

Watch the video explanation below:

Given

f(x) = 6x³ + √2x² − 10x − 4√2

One zero of the polynomial is:

x = √2

To Find

The other two zeroes of the given polynomial.

Solution

Step 1: Use Factor Theorem

Since √2 is a zero of the polynomial,

(x − √2) is a factor of f(x).

Step 2: Divide f(x) by (x − √2)

Divide 6x³ + √2x² − 10x − 4√2 by (x − √2).

First term:

6x³ ÷ x = 6x²

Multiply:

6x²(x − √2) = 6x³ − 6√2x²

Subtract:

(6x³ + √2x²) − (6x³ − 6√2x²)

= 7√2x²

Bring down −10x:

7√2x² − 10x

Next term:

7√2x² ÷ x = 7√2x

Multiply:

7√2x(x − √2) = 7√2x² − 14x

Subtract:

(7√2x² − 10x) − (7√2x² − 14x)

= 4x

Bring down −4√2:

4x − 4√2

Next term:

4x ÷ x = 4

Multiply:

4(x − √2) = 4x − 4√2

Subtract:

(4x − 4√2) − (4x − 4√2) = 0

So remainder is zero.

Quotient obtained:

6x² + 7√2x + 4

Step 3: Find the Remaining Zeroes

Now solve:

6x² + 7√2x + 4 = 0

Split the middle term:

6x² + 3√2x + 4√2x + 4 = 0

3x(2x + √2) + 2√2(2x + √2) = 0

(2x + √2)(3x + 2√2) = 0

∴ 2x + √2 = 0 or 3x + 2√2 = 0

∴ x = −√2/2 or x = −2√2/3

Final Answer

The three zeroes of the given polynomial are:

√2, −√2/2 and −2√2/3

Conclusion

Hence, the cubic polynomial 6x³ + √2x² − 10x − 4√2 has zeroes √2, −√2/2 and −2√2/3.