Finding a polynomial of a given degree with given zeros: Real...
Find a polynomial $f(x)$ of degree 5 that has the following zeros.
7 (multiplicity 2 ), $-5,-1,0$
Leave your answer in factored form.
\(\boxed{f(x) = x(x - 7)^2(x + 1)(x + 5)}\)
Step 1 :The problem is asking for a polynomial of degree 5 with given zeros. The zeros are 7 (with multiplicity 2), -5, -1, and 0.
Step 2 :A polynomial with given zeros can be found by multiplying factors of the form \((x - a)\), where \(a\) is a zero of the polynomial. If a zero has a multiplicity of \(n\), then the factor \((x - a)\) should be raised to the power of \(n\).
Step 3 :In this case, the polynomial can be found by multiplying the factors \((x - 7)^2\), \((x + 5)\), \((x + 1)\), and \(x\).
Step 4 :\(f(x) = x(x - 7)^2(x + 1)(x + 5)\)
Step 5 :\(\boxed{f(x) = x(x - 7)^2(x + 1)(x + 5)}\)