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If a is a zero of f(x)=0, then (x-a) is a factor of f(x).
f(x)=x^4-5x^3+5x^2+5x-6
We want to write a polynomial function with integral coefficients so that f(x)=0 has the given zeros.
3, -1, 1, 2
Recall that if a is a zero of f(x)=0, then (x-a) is a factor of f(x).
| Root | Factor |
|---|---|
| 3 | x-3 |
| -1 | x-(-1) |
| 1 | x-1 |
| 2 | x-2 |
| Polynomial | f(x)= (x-3) [x-(-1)] (x-1)(x-2) |
Let's simplify the polynomial by applying the Distributive Property. For simplicity, we will start by multiplying the first two factors and the last two factors separately. (x-3) & * [x-(-1)] (x-1) & * (x-2) After we find these products, we will multiply the obtained expressions.
Let's continue by finding the product of the last two factors.
Finding the product of these two polynomials will give us the desired polynomial function.
Distribute (x^2-3x+2)
Distribute x^2
Distribute - 2x
Distribute -3
Add and subtract terms