Question

Find other zeros of the polynomial
$p(x) = {2x}^{4 } + {7x}^{3} - {19x}^{2} - 14x + 30$
if two of its zeros are
$\sqrt{2} \: and - \sqrt{2}$
​

Answer 1

SOLUTION:-

Given:

p(x)= 2x⁴ + 7x³ -19x²-14x+30.

Since, two zeroes are √2 & -√2.

Therefore,

(x-√2)(x+√2) = x² -2 is a factor of the given polynomial.

Now,

We divide the given polynomial by x² -2.

So,

Above in attachment.

Therefore,

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

Now,

2x² +7x-15=0

=) 2x² +10x-3x -15=0

=) 2x(x+5) -3(x+5)=0

=) (x+5)(2x-3)=0

=) x+5= 0 or 2x -3 = 0

=) x= -5 or 2x = 3

=) x= -5 or x= 3/2

So,

It's zeroes are given by x= -5 & x= 3/2.

Hence,

The zeroes of the given polynomial are;

√2, (-√2), (-5) & 3/2

Hope it helps ☺️