Question

you have to answer both question
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Answer 1

Step-by-step explanation:

(i)

Given p(x) = x⁴ - 6x³ - 26x² - 138x - 35.

Given zeroes are (2 + √3) and (2 - √3).

∴ [x - (2 + √3)][(x - (2 - √3)]

= (x - 2 - √3)(x - 2 + √3)

= (x - 2)² - (√3)²

= x² + 4 - 4x - 3

= x² - 4x + 1

So, x² - 4x + 1 is a factor of p(x).

Now,

We have to divide p(x) by x² - 4x + 1.

Long Division Method:

x² - 4x + 1) x⁴ - 6x³ - 26x² + 138x - 35 (x² - 2x - 35

                 x⁴ - 4x³ + x²

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

                        -2x³ - 27x² + 138x

                         -2x³ + 8x² - 2x

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

                                     -35x² + 140x - 35

                                      -35x² + 140x - 35

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

                                                          0

Now,

We have to find other two zeroes of x² - 2x - 35.

= x² - 7x + 5x - 35

= x(x - 7) + 5(x - 7)

= (x + 5)(x - 7)

So, its zeroes are -5, 7.

Hence, all the zeroes of the polynomial are:

2 + √3, 2 - √3, -5,7

(ii)

Given polynomial is 3x² - 8x + 2k + 1.

Let one zero of the polynomial is α.

Then, the other zero will be 7α.

Sum of zeroes:

α + 7α = -b/a

⇒ 8α = -8/3

⇒ α = -1/3

Product of zeroes:

α * 7α = c/a

⇒ 7α² = (2k + 1)/3

⇒ 7(-1/3)² = (2k + 1)/3

⇒ 7(1/9) = (2k + 1)/3

⇒ 7/3 = (2k + 1)

⇒ 4/3 = 2k

⇒ 4/6 = k

⇒ k = 2/3.

∴ Thus, value of k = 2/3.

Hope it helps!

Answer 2

hope it helps!..................