Answer:
To create a polynomial that is factorable and can be factored by grouping, we can start with a polynomial of degree 3 in factored form. Let's use the following conditions:
- The polynomial has a zero at x = 2.
- The polynomial has a zero at x = -1.
- The polynomial has a zero at x = 4.
To satisfy these conditions, we can start by writing the factors of the polynomial:
(x - 2)(x + 1)(x - 4)
Now, we can multiply these factors together to get the polynomial:
(x - 2)(x + 1)(x - 4) = (x^2 - x - 2)(x - 4)
= x^3 - 4x^2 - x^2 + 4x - 2x + 8
= x^3 - 5x^2 + 2x + 8
So, the polynomial of degree 3 that satisfies the given conditions and can be factored by grouping is:
f(x) = x^3 - 5x^2 + 2x + 8
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