Question

Find all zeros of the polynomial2x⁴+7x³-19x²-14x+30, if two of its zeros are √2 and –√2.

Answer 1

x = 3/2 and x = -5.

Step-by-step explanation:

Polynomial f(x) = 2x⁴+7x³-19x²-14x+30

√2 and –√2. are zeroes of the polynomial.

Then (x+√2) and (x-√2) are factors of the polynomial and they divide the polynomial.

So (x+√2)*(x-√2) = x²+2

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

Now let's factorize 2x² + 7x -15

2x² + 7x -15 = 0

2x² +10x -3x -15 = 0

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

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

So the other roots are x = 3/2 and x = -5.

Answer 2

x = 3/2 and x = -5.

Step-by-step explanation:

Polynomial f(x) = 2x⁴+7x³-19x²-14x+30

√2 and –√2. are zeroes of the polynomial.

Then (x+√2) and (x-√2) are factors of the polynomial and they divide the polynomial.

So (x+√2)*(x-√2) = x²+2

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

Now let's factorize 2x² + 7x -15

2x² + 7x -15 = 0

2x² +10x -3x -15 = 0

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

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

So the other roots are x = 3/2 and x = -5.