Which of the following is a polynomial function in factored from with zeros at -6,-2, and 3?

Answer 1

The short answer: A polynomial in factored form has all of its zeroes in plain sight. An example of a factored polynomial with zeroes at -6, -2, and 3 is the polynomial 4(x + 6)(x + 2)(x - 3). The zeroes are found by setting each binomial factor equal to zero and solving for x. This works because of the zero-product rule: If one of the binomial factors in the factored form is equal to zero, then the whole polynomial is equal to zero. Notice that x = -6 makes x + 6 = 0, and x = -2 makes x + 2 = 0, and x = 3 makes x - 3 = 0. Also notice that the constant factor 4 at the beginning of the polynomial does not affect the value of the zeroes at all.

The long answer: First, what are "zeroes"? Well, recall that a polynomial (in high-school algebra terms) is just a sum of various terms involving powers of x. For example, consider the polynomial x² - x - 2. One interesting question is, what values can we plug in for x to make the value zero? Any number that we can plug into x to make the result zero is called a "zero" of the polynomial. It turns out that the only numbers that work are x = 2 or x = -1. (Observe that 2² - 2 - 2 = 0 and (-1)² + 1 - 2 = 0.) But that those are the solutions is not necessarily obvious from just looking at the polynomial.

If, however, you notice that x² - x - 2 can be rewritten as the product of two binomials, (x - 2) • (x + 1), then the solution is much easier. Now the equation can be written (x - 2)(x + 1) = 0. When you multiply two or more numbers together and the result is zero, it must be true that at least one of the numbers you multiplied was zero. Here, we're multiplying (x - 2) and (x + 1) and saying that the result is equal to zero; therefore, either (x - 2) or (x + 1) must be zero. If x - 2 = 0, then x = 2. Alternatively, if x + 1 = 0, then x = -1. This means that 2 and -1 are the zeroes of the original polynomial, x² - x - 2.

The expression (x - 2)(x + 1) is the factored form of the polynomial x² - x - 2. The factored form is useful because you can identify the zeroes of the polynomial simply by setting each factor to zero and solving for x.

To circle back to the original question, here are some examples of polynomials that have -6, -2, and 3 as zeroes. The first two examples are in completely factored form, but the third example is not:

1. (x + 6)(x + 2)(x - 3)
2. 2(x + 6)(x + 2)(x - 3)
3. Not in factored form: (x + 6)(2x + 4)(x - 3)

The first example is as simple as it gets. x = -6 makes the first factor zero, x = -2 makes the second factor zero, and x = 3 makes the third factor zero.

The second example demonstrates that multiplying the entire polynomial by a constant doesn't change the zeroes.

The third example is like the second example, but the 2 has been distributed into the (x + 2) factor. This serves as a warning not to just choose the second term in each binomial to be the zero as a shortcut. Make sure you are setting each factor to zero and solving for x. (i.e., 2x + 4 = 0 means that x = -2)