Problem 70 - College Algebra with Corequisite Support

Question

For the following exercises, construct a polynomial function of least degree possible using the given information. Real roots: -4,-1,1,4 and \((-2, f(-2))=(-2,10)\)

Step-by-Step Solution

Answer

The polynomial is \(f(x) = \frac{5}{18}(x + 4)(x + 1)(x - 1)(x - 4)\).

Step 1: List Given Real Roots

The real roots provided are \(-4, -1, 1,\) and \(4\). These roots suggest possible factors of the polynomial function.

Step 2: Construct Polynomial from Roots

Since \(f(x)\) is a polynomial with the given roots, it can be expressed as:\[f(x) = a(x + 4)(x + 1)(x - 1)(x - 4)\] where \(a\) is a constant coefficient to be determined.

Step 3: Use the Point to Determine 'a'

We know that the function passes through the point \((-2, 10)\). Substitute \(x = -2\) and \(f(-2) = 10\) into the polynomial:\[10 = a(-2 + 4)(-2 + 1)(-2 - 1)(-2 - 4)\]This simplifies to \[10 = a(2)(-1)(-3)(-6)\].

Step 4: Simplify and Solve for 'a'

Solve the equation from Step 3:\[10 = a(2)(-1)(-3)(-6)\]Calculate the product:\[10 = a(36)\]Thus,\[a = \frac{10}{36} = \frac{5}{18}\].

Step 5: Write the Final Polynomial

Substitute \(a = \frac{5}{18}\) back into the polynomial:\[f(x) = \frac{5}{18}(x + 4)(x + 1)(x - 1)(x - 4)\]This is the polynomial function of least degree that satisfies the given conditions.

Key Concepts

Real Roots

In the realm of polynomial functions, real roots are values of the variable that make the entire polynomial equal to zero. When you're asked to construct a polynomial from real roots, these roots serve as your building blocks. For instance, if you have real roots such as -4, -1, 1, and 4, each of these values translates into a factor of the polynomial.

For any real root 'r', \(x - r\) becomes a factor of the polynomial. This means for the roots given:

• -4 becomes \(x + 4\)
• -1 becomes \(x + 1\)
• 1 becomes \(x - 1\)
• 4 becomes \(x - 4\)

By multiplying these factors, you construct a polynomial function that inherently includes these roots.

Constant Coefficient

The constant coefficient 'a' in a polynomial, like in \(f(x) = a(x + 4)(x + 1)(x - 1)(x - 4)\), is pivotal in defining the size of the polynomial's output without altering its roots. This coefficient impacts the vertical stretch or compression of the graph of the function.

To find this constant accurately, you need additional information—usually a specific point through which the graph passes. For example, if the polynomial passes through \((-2, 10)\), you substitute \(-2\) into the expression for \(x\) and solve for 'a' using the equation \ f(x) . This ensures the polynomial exactly meets the condition at that point.

After solving, you may find a fraction or whole number for 'a'. This value becomes the multiplier for all factors containing the roots, ensuring that the polynomial behaves correctly across its domain.

Factorization

Factorization in the context of polynomials involves breaking down the polynomial into simpler terms or factors that, when multiplied, reconstruct the original polynomial. Each factor corresponds to the expression \(x - r\), where \(r\) represents a known root.

This method simplifies analysis and offers a clear visual of the function's roots. For instance, a polynomial derived from roots -4, -1, 1, and 4 would look something like \(f(x) = a(x + 4)(x + 1)(x - 1)(x - 4)\). All these factors, when expanded, lead back to the polynomial's full form.

Factorization is an essential skill in algebra, offering insights into how polynomials behave, including the points where they cross or touch the x-axis. It's particularly useful in solving polynomial equations, allowing a step-by-step pathway to find each solution or root.

Degree of Polynomial

The degree of a polynomial is crucial as it indicates the highest power of the variable present in the polynomial. This degree tells us about the polynomial's most significant term, guiding us on how the function grows and behaves for large values of the variable.

For a polynomial constructed from given real roots using factors like \( (x + 4)(x + 1)(x - 1)(x - 4)\), the degree is determined by counting these factors. You simply look at the number of terms being multiplied.

Here, we have four roots, meaning the polynomial has four linear factors. Thus, its overall degree is 4, which is a critical piece of information when graphing or solving polynomials as it dictates the curve's shape and the number of possible real roots. A polynomial's degree offers a peek into its complexity and intricacy, defining how many times a graph can potentially cross the x-axis.