Problem 285 - Precalculus

Question

For the following exercises, use the given information about the polynomial graph to write the equation. Degree \(3 .\) Zeros at \(x=-3, \quad x=-2\) and \(x=1 . y\) -intercept at \((0,12)\)

Step-by-Step Solution

Answer

The polynomial equation is \(P(x) = -2(x + 3)(x + 2)(x - 1)\).

Step 1: Identify Polynomial Structure

Since the polynomial has three roots at \(x = -3\), \(x = -2\), and \(x = 1\), and degree 3, we can express the polynomial in factored form as \(P(x) = a(x + 3)(x + 2)(x - 1)\). Here, \(a\) is a constant coefficient that we'll determine next.

Step 2: Use y-intercept to Determine 'a'

The polynomial passes through the point \((0, 12)\). Substitute \(x = 0\) and \(y = 12\) into the equation \(P(x) = a(x + 3)(x + 2)(x - 1)\). This gives us: \[P(0) = a(0 + 3)(0 + 2)(0 - 1) = 12\]\[a(3)(2)(-1) = 12\]\[-6a = 12\]Solving for \(a\), we get \(a = \frac{-12}{6} = -2\).

Step 3: Write the Final Polynomial Equation

Substitute \(a = -2\) back into the polynomial equation:\[P(x) = -2(x + 3)(x + 2)(x - 1)\]This is the equation of the polynomial with the given features.

Key Concepts

Polynomial Roots

The concept of polynomial roots is central to understanding polynomial equations. Roots, also known as zeros, are the values of \(x\) for which the polynomial equation evaluates to zero. In other words, if you have a polynomial \(P(x)\), a root \(r\) is a solution such that \(P(r) = 0\). For the given exercise, the roots of the polynomial are \(x = -3\), \(x = -2\), and \(x = 1\). Each of these roots corresponds to a point where the polynomial graph touches or crosses the \(x\)-axis. The degree of the polynomial equals the number of roots if all roots are distinct, as in this example. Understanding roots is crucial because they provide the backbone for constructing the polynomial equation in its factored form.

Degree of Polynomial

The degree of a polynomial is a key feature that tells us about its complexity and behavior. The degree is the highest power of \(x\) in the polynomial equation when it is fully expanded. For instance, a polynomial \(P(x)\) with degree 3 will typically have terms up to \(x^3\). In the exercise, we are given a polynomial of degree 3. This implies it can have up to three roots, as each degree corresponds to one potential root. The degree also tells us about the number of turning points and the end behavior of the graph. A degree 3 polynomial will have a graph that can have up to 2 turning points, and its end behavior will reflect that of the leading term, which in this simplified example is \(-2x^3\), thus heading towards negative or positive infinity as \(x\) moves towards infinity or negative infinity.

Polynomial Intercept

Intercepts are important features that provide specific points through which the polynomial graph passes. The \(y\)-intercept is the point where the graph intersects the \(y\)-axis. To find it, we evaluate the polynomial at \(x = 0\). In this exercise, the \(y\)-intercept is given as \((0, 12)\), meaning when \(x = 0\), \(y\) is 12. Intercepts like these are useful for checking or determining coefficients in the polynomial equation. In the solution step, this intercept was used to find the constant \(a\) in the polynomial's equation, confirming the full polynomial once plugged into the factorized form of \( -2(x + 3)(x + 2)(x - 1)\). Intercepts help ensure the accuracy of the equation by serving as a validation point.

Factored Form of Polynomial

The factored form of a polynomial is a way of expressing the polynomial as a product of its linear factors corresponding to its roots. Each root gives a factor of the form \((x - r)\), where \(r\) is a root of the polynomial. Given the roots \(-3\), \(-2\), and \(1\) in the exercise, the polynomial in factored form starts off as \((x + 3)(x + 2)(x - 1)\). The additional step is to determine a constant factor, \(a\), which adjusts the scale of the polynomial to match any additional criteria such as intercepts. For this exercise, the factored form including the constant is \(-2(x + 3)(x + 2)(x - 1)\). Factored form is extremely useful not just for constructing equations but also for simplifying the understanding of where the graph intersects the \(x\)-axis.