4. A construction worker needs to put
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It is given in the question that
And
can also be written as
Therefore,
That gives,
Required polynomial of degree four having zeros as -2 , 4 , 4 , 8 is
Need to determine a polynomial of degree 4 and the zeros are -2, 4 , 4 and 8
Let the required polynomial be represented by f(x)
The factor theorem describes the relationship between the root of a polynomial and a factor of the polynomial.
If the polynomial p(x) is divided by cx−d and the remainder, given by p(d/c), is equal to zero, then cx−d is a factor of p(x).
-2 is zero of a polynomial means when x = -2, f(-2) = 0, so from factor theorem we can say that
=> x = -2 that is x + 2 = 0 is factor of polynomial f(x)
4 is zero of a polynomial means when x = 4, f(4) = 0 , so from factor theorem we can say that
=> x = 4 that is x -4 = 0 is factor of polynomial f(x)
4 is zero of a polynomial means when x = 4, f(4) = 0 , so from factor theorem we can say that
=> x = 4 that is x -4 = 0 is factor of polynomial f(x)
8 is zero of a polynomial means when x = 8, f(8) = 0 , so from factor theorem we can say that
=> x = 8 that is x -8 = 0 is factor of polynomial f(x)
So now we have four factors of polynomial f(x) that are (x + 2), (x -4) , (x -4) and (x – 8)
And as given that degree of polynomial f(x) is 4
Now f(x) is equal to product of factors
Hence required polynomial of degree four having zeros as -2 , 4 , 4 , 8 is