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For the polynomial below, 2 is a zero. f(x) = x2 + 2x² – 7x – 2 Express f(x) as a product of linear factors.​

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Final answer:

To express the polynomial f(x) = x^2 + 2x² - 7x - 2 as a product of linear factors, we need to find the other zero of the polynomial. The polynomial can be factored as (x - 2)(x + 1.5 - 0.866i)(x + 1.5 + 0.866i).

Step-by-step explanation:

To express the polynomial f(x) = x^2 + 2x² - 7x - 2 as a product of linear factors, we need to find the other zero of the polynomial. Since 2 is a zero, we can use synthetic division to factor out (x - 2). This gives us a new polynomial, x^2 + 3x + 1.

Now, we can use the quadratic formula or factoring to find the remaining zeros of the new polynomial. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using:

x = (-b +- sqrt(b^2 - 4ac)) / (2a)

By applying the quadratic formula to the polynomial x^2 + 3x + 1, we find two complex solutions: x ≈ -1.5 + 0.866i and x ≈ -1.5 - 0.866i.

Therefore, f(x) can be expressed as a product of linear factors: f(x) = (x - 2)(x + 1.5 - 0.866i)(x + 1.5 + 0.866i).

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