Problem 64 - College Algebra

Question

For the following exercises, use the given information about the polynomial graph to write the equation. Degree \(5 .\) Roots of multiplicity 2 at \(x=-3\) and \(x=2\) and a root of multiplicity 1 at \(x=-2 . y\) -intercept at \((0,4)\)

Step-by-Step Solution

Answer

The polynomial is \( p(x) = \frac{1}{18}(x + 3)^2 (x - 2)^2 (x + 2) \).

Step 1: Identify the Roots and Their Multiplicities

The polynomial is of degree 5 and has roots at the following points with specified multiplicities:1. Root at \(x = -3\) with multiplicity 2.2. Root at \(x = 2\) with multiplicity 2.3. Root at \(x = -2\) with multiplicity 1.

Step 2: Write the General Form of the Polynomial

Using the roots identified, the polynomial can be expressed as:\[p(x) = a(x + 3)^2 (x - 2)^2 (x + 2)\]Here, \(a\) is a constant that will be determined using additional information provided.

Step 3: Substitute the y-Intercept to Find the Constant

The y-intercept is given as \((0, 4)\), meaning when \(x = 0\), \(p(x) = 4\).Substitute in the equation:\[p(0) = a(0 + 3)^2 (0 - 2)^2 (0 + 2) = 4\]Simplify:\[a(9)(4)(2) = 4\]\[72a = 4\]

Step 4: Solve for the Constant

Solve the equation \(72a = 4\) to find \(a\):\[a = \frac{4}{72} = \frac{1}{18}\]

Step 5: Write the Final Polynomial Equation

Substitute the value of \(a\) back into the polynomial equation:\[p(x) = \frac{1}{18}(x + 3)^2 (x - 2)^2 (x + 2)\]This is the equation of the polynomial.

Key Concepts

Roots and Multiplicity

A polynomial equation's roots are the values of \( x \) that make the polynomial zero. In other words, they are the solutions to the equation \( p(x) = 0 \). Each root can have a multiplicity, which indicates how many times that particular root appears as a factor in the polynomial. A higher multiplicity means the polynomial touches or crosses the x-axis at the root without changing direction sharply.

For instance, in our example polynomial, we have roots at \( x = -3 \), \( x = 2 \), and \( x = -2 \). The root \( x = -3 \) has a multiplicity of 2, indicating it appears twice in the factorization. Similarly, \( x = 2 \) also has a multiplicity of 2. The root \( x = -2 \) has a multiplicity of 1, which means it appears once.

The polynomial equation using these roots can be expressed through factors: \((x + 3)^2\) for the root \( x = -3 \), \((x - 2)^2\) for the root \( x = 2 \), and \((x + 2)\) for the root \( x = -2 \). This multiplicity explains the number of times each factor repeats in the polynomial equation.

Degree of Polynomial

The degree of a polynomial tells us about the highest power of \( x \) in the polynomial. It also gives insight into the polynomial's behavior as \( x \) approaches infinity, and the number of roots, counting multiplicities, that it can have. The given polynomial is of degree 5, which means that the sum of the multiplicities of the roots must equal 5.

In our specific polynomial:

• The root \( x = -3 \) with multiplicity 2 contributes \( 2 \) to the degree.
• The root \( x = 2 \) with multiplicity 2 also adds \( 2 \) to the total degree.
• The root \( x = -2 \) with multiplicity 1 contributes \( 1 \) to the degree.

Adding these multiplicities together gives \( 2 + 2 + 1 = 5 \), which confirms that the polynomial is indeed of degree 5. A degree 5 polynomial generally has a maximum of 5 zero-crossings and 4 turning points.

Y-intercept

The y-intercept of a polynomial is the point where the graph of the polynomial crosses the y-axis. This occurs when \( x = 0 \). Therefore, finding the y-intercept involves calculating \( p(0) \).

In our example, the y-intercept is given as \( (0, 4) \). This means that when \( x = 0 \), the value of the polynomial is 4. We used this information to solve for the constant \( a \) in the polynomial. By substituting \( x = 0 \) into the equation \( p(x) = a(x + 3)^2 (x - 2)^2 (x + 2) \), and adjusting \( a \) so that the expression equals 4, we determined the value of the constant that completes the equation.

Knowing the y-intercept is crucial because it helps in graphing the polynomial and verifying the overall correctness of the polynomial equation.

Solving Polynomial Equations

Solving polynomial equations involves finding all the roots or the values of \( x \) that satisfy \( p(x) = 0 \). In our example, the polynomial equation was constructed based on the given roots and their multiplicities from the polynomial's graph.

To form the equation, we used:

• The root \( x = -3 \) with multiplicity 2 gave us the factor \((x + 3)^2\).
• The root \( x = 2 \) with multiplicity 2 gave us \((x - 2)^2\).
• The root \( x = -2 \) with multiplicity 1 gave us \((x + 2)\).

After laying out these factors, we introduced a constant \( a \) to account for any vertical stretching or compression of the polynomial's graph. Using the given y-intercept, we then solved for \( a \), ensuring that the polynomial equation reflects the y-intercept correctly.

In practice, solving polynomial equations may involve techniques such as factoring, using the quadratic formula for degree 2 equations, or numerical methods for higher degrees if explicit factoring is not feasible.