Find all the zeroes of the polynomial 2x 4 7x 19x2 14x + 30, if two of its zeroes are

Text Solution

Solution : Given `p(x)=2x^4+7x^3−19x^2−14x+30`<br> Two of its zeroes are `sqrt(2) ​`and `−sqrt(2)`<br> ​so,`g(x)=(x−sqrt(2))(x+sqrt(2))`<br> or,`x^2−2`<br> so we get <br>,divisor=`x^2−2`<br> Quotient=`2x^2+7x−15`<br> Remainder=`0`<br> so,`q(x)=−2x^2+7x−15`<br> `2x^2+10x−3x−15=0`<br> =`2x(x+5)−3(x+5)=0`<br> =`(2x−3)(x+5)=0`<br> thus,`x=3/2,-5`<br>​so,roots of equation is `sqrt(2) ​,−sqrt(2) ​,3/2 ​` and `−5`

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Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and coefficients in each case:
2x3 + x2 –5x + 2; 1/2, 1,–2 

Let p(x) = 2x3+x2–5x + 2 Comparing the given polynomial with ax3 + bx3 

+ cx + d, we get