Question

obtain all the zeroes of 3x⁴ - 15x³ + 13x² + 25x - 30, if two of its zeroes are √5/3 and -√5/3.​

Answer 1

All the zeroes of $\bf 3 {x}^{4} - 15 {x}^{3} + 13 {x}^{2} + 25x - 30$ are $\bf 2, \: 3, \: \sqrt{ \frac{5}{3} }\: and \: -\sqrt{ \frac{5}{3} }$

Given:

• A quartic polynomial.
• $3 {x}^{4} - 15 {x}^{3} + 13 {x}^{2} + 25x - 30$
• Two of the zeroes are √(5/3) and -√(5/3).

To find:

• Find all zeroes of quartic polynomial.

Solution:

Step 1:

Obtain factors of quartic polynomial from given zeros.

As zeros are

$x = \sqrt{ \frac{5}{3} } \: and \: x = - \sqrt{ \frac{5}{3} }$

so,

factors are

$(x - \sqrt{ \frac{5}{3} } ) \: and \: (x + \sqrt{ \frac{5}{3} } )$

Step 2:

Obtain quadratic polynomial by multiplying two factors.

As

$\bf (a - b)(a + b) = {a}^{2} - {b}^{2}$

so,

$(x - \sqrt{ \frac{5}{3} } )(x + \sqrt{ \frac{5}{3} } ) = {x}^{2} - \frac{5}{3}$

Thus,

The quadratic equation which is factor of given quartic polynomial is

$3 {x}^{2} - 5=0$

Step 3:

Divide the given quartic polynomial by 3x²-5.

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 3 {x}^{2} - 5 \: ) \: 3 {x}^{4} - 15 {x}^{3} + 13 {x}^{2} + 25x - 30( \: {x}^{2} - 5x + 6 \\ 3 {x}^{4} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: - 5 {x}^{2} \\ - - - - - - - - - \\ \: \: \: \: \: \: \: \: - 15 {x}^{3} + 18 {x}^{2} + 25x - 30 \\ - 15 {x}^{3} \: \: \: \: \: \: \: \: \: \: + 25x \\ - - - - - - - - - - - \\ 18 {x}^{2} - 30 \\ 18 {x}^{2} - 30 \\ - - - - - - \\ \times \times \\ - - - - - -$

Step 4:

Factorise the quotient polynomial.

${x}^{2} - 5x + 6$

or

${x}^{2} -2x-3x+6$

or

${x}(x-2) -3(x-2)$

or

$(x-2)(x-3)$

Thus,

Zeroes are

$(x - 2)(x - 3) = 0$

or

$\bf x = 2$

and

$\bf x = 3$

Thus,

All zeroes are 2,3,√(5/3) and -√(5/3).

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