Question

find all other zeroes of 2x⁴+7x³-19x²-14x+30
it two of its zeroes
are √2and-√2.​

Answer 1

Solution:

Given Polynomial –

→ p(x) = 2x⁴ + 7x³ - 19x² - 14x + 30

Two of its roots are √2 and -√2

So,

→ x = √2

→ x - √2 = 0

→ x - √2 is a factor.

Again,

→ x = -√2

→ x + √2 = 0

→ x + √2 is a factor.

So,

→ (x + √2)(x - √2) is also a factor.

→ x² - 2 is a factor.

Now, divide p(x) by x² - 2

x² - 2 ) 2x⁴ + 7x³ - 19x² - 14x + 30 ( 2x² + 7x - 15

           2x⁴          -  4x²

        –                +

———————————————————

                    7x³ - 15x² - 14x + 30

                    7x³           - 14x²

                 –                 +

———————————————————

                            -15x² + 30

                            -15x² + 30

———————————————————

                                     0

So,

→ p(x) = (x² - 2)(2x² - 7x + 5)

Now, factorise 2x² - 7x + 5

2x² - 7x + 5

= 2x² - 2x - 5x + 5

= 2x(x - 1) - 5(x - 1)

= (2x - 5)(x - 1)

Therefore,

→ (2x - 5)(x - 1) = 0

→ 2x - 5 = 0 or x - 1 = 0

→ x = 2½ and 1

• So, the other roots are 2½ and 1.

Answer:

• The other two roots are 2½ and 1.