Given that x − √5 is a factor of the cubic polynomial x³ − 3√5x² + 13x − 3√5, find all the zeroes of the polynomial

Video Explanation

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Given

f(x) = x³ − 3√5x² + 13x − 3√5

One factor of the polynomial is:

x − √5

To Find

All the zeroes of the given polynomial.

Solution

Step 1: Use Factor Theorem

Since x − √5 is a factor of the polynomial, therefore

x = √5 is a zero of f(x).

Step 2: Divide f(x) by (x − √5)

Divide x³ − 3√5x² + 13x − 3√5 by (x − √5):

First term:

x³ ÷ x = x²

Multiply:

x²(x − √5) = x³ − √5x²

Subtract:

(x³ − 3√5x²) − (x³ − √5x²)

= −2√5x²

Bring down +13x:

−2√5x² + 13x

Next term:

−2√5x² ÷ x = −2√5x

Multiply:

−2√5x(x − √5) = −2√5x² + 10x

Subtract:

(−2√5x² + 13x) − (−2√5x² + 10x)

= 3x

Bring down −3√5:

3x − 3√5

Next term:

3x ÷ x = 3

Multiply:

3(x − √5) = 3x − 3√5

Subtract:

(3x − 3√5) − (3x − 3√5) = 0

So remainder is zero.

Quotient obtained:

x² − 2√5x + 3

Step 3: Find the Remaining Zeroes

Now solve the quadratic equation:

x² − 2√5x + 3 = 0

Split the middle term:

x² − √5x − √5x + 3 = 0

x(x − √5) − √5(x − √5) = 0

(x − √5)(x − √5) = 0

∴ x = √5 (repeated root)

Final Answer

All the zeroes of the given polynomial are:

√5, √5 and √5

Conclusion

Hence, the cubic polynomial x³ − 3√5x² + 13x − 3√5 has all three zeroes equal to √5.