Question

Obtain all zeroes of the polynomial, if two of its zeroes are -root 5 and root 5​

Answer 1

Step-by-step explanation:

Let p(x) = 2x⁴ - x³ - 11x² + 5x + 5.

Since x = √5 is a zero, x - √5 is a factor.

Since x = -√5 is a zero, x + √5 is a factor.

Hence,

(x - √5)(x + √5) is a factor.

⇒ x² - 5 is a factor.

Now, By dividing the given polynomial by x² - 5.

We can find out other factors.

Long Division Method:

x² - 5) 2x⁴ - x³ - 11x² + 5x + 5 (2x² - x - 1

          2x⁴       - 10x²

          ---------------------------------

                  - x³   - x² + 5x + 5

                  - x³          + 5x

 

          ----------------------------------

                             - x²         +  5

                              - x²        +  5

            ----------------------------------

                                             0

Now,

We factorize 2x² - x - 1

= 2x² - 2x - x - 1

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

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

= (2x - 1)(x - 1) = 0

x = 1, 1/2

∴ x = (1/2) & 1 are the zeroes of p(x).

Therefore, the zeroes of p(x) are √5, -√5, (1/2), 1

Hope it helps!

Answer 2

here is your answer mate .

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