Find a polynomial of degree n that has only the given zero(s) x = 5, -1 and degree n = 4.
Polynomials are very important concepts that are related to algebra. A polynomial can have one or more variables to it as well as any value of its degree. The degree of a polynomial is the highest power of a variable which is the part of the polynomial. Let's solve an interesting problem related to degrees and polynomials.
Answer: The polynomial of degree n = 4, and zero(s) x = 5, -1, is x4 - 8x3 + 6x2 + 40x + 25.
Let's understand the solution deeply.
The polynomial of degree 4 is called a biquadratic polynomial.
Also, the given number of zeroes are 5 and -1, but the degree is 4. So, the polynomial can't have all unique zeros.
Hence, let the multiplicity of each of the two zeroes be 2.
Therefore, the polynomial can be f(x) = (x - 5)2 (x + 1)2 = x4 - 8x3 + 6x2 + 40x + 25
From above, we can see that the polynomial f(x) has the highest power of x as 4, hence it has a degree of 4.