Question

Find all the zeroes of the polynomial \[2{x^4} - 11{x^3} + 7{x^2} + 13x - 7\] it being given that two of its zeroes are \[(3 + \sqrt 2 )\] and \[(3 - \sqrt 2 )\] .

Answer 1

Hint: As we see that we have been given two zeroes of the polynomial. Zeroes are the roots of the polynomial. With our previous knowledge we all see that the complex roots are occurring in pairs. So, now we factorize the polynomial using the roots.

Complete step-by-step solution:
Let \[f(x) = 2{x^4} - 11{x^3} + 7{x^2} + 13x - 7\] be our given polynomial function whose roots or zeros \[(3 + \sqrt 2 )\] and \[(3 - \sqrt 2 )\] are given to us.
As \[(3 + \sqrt 2 )\] and \[(3 - \sqrt 2 )\] are roots of the equation we have that \[(x - (3 + \sqrt 2 ))\] and \[(x - (3 - \sqrt 2 ))\] are two factors of the polynomial\[f(x)\] .
Then \[(x - (3 - \sqrt 2 )).(x - (3 + \sqrt 2 ))\] is a factor of the polynomial function, which implies \[{x^2} - 6x + 7\] is a factor of the polynomial function.
Now let us divide the polynomial function by \[{x^2} - 6x + 7\] to get its remaining factor,
So, we found that the other factor is \[2{x^2} + x - 1\] and hence,
\[(x - (3 - \sqrt 2 )).(x - (3 + \sqrt 2 )).(2{x^2} + x - 1) = 0\]
\[
   \Rightarrow (x - (3 - \sqrt 2 )).(x - (3 + \sqrt 2 )).(2x - 1)(x + 1) = 0 \\
   \Rightarrow x = - 1,\dfrac{1}{2},(3 - \sqrt 2 ),(3 + \sqrt 2 ) \\
 \]
Hence we find that the other two factors of the given polynomial equation \[f(x) = 2{x^4} - 11{x^3} + 7{x^2} + 13x - 7\] are\[ - 1,\dfrac{1}{2}\] .
Additional information: If we split down the word polynomial, we get “Poly” and “Nominal”. Poly stands for many and nominal implies the term and as a result while they are mixed we will say that polynomials are algebraic expressions with many terms. Polynomial functions are taken into consideration to be the most effective, mist normally used, and most important mathematical functions. These functions are often used in the actual world and are taken into consideration to be the constructing blocks of Algebra. Polynomial functions also cover a wide quantity of different types. It is critical for one to look at and understand polynomial features due to their vast and wide variety.

Note: It is important that we understand that zeroes and roots of a function are the same thing. It is also very important that we know that the degree of the function tells us the number of roots it has and then we use the given conditions to factorize the function and search for the other zeroes.