Problem 70 - College Algebra

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 \(P(x) = -\frac{5}{18}(x^4 - 17x^2 + 16)\).

Step 1: Identify the Roots

The polynomial has real roots at \(-4, -1, 1,\) and \(4\). These roots indicate that the polynomial can be written as the product of factors: \((x + 4)(x + 1)(x - 1)(x - 4)\).

Step 2: Determine the Polynomial in Factored Form

Since the polynomial has the real roots identified in Step 1, the function can be expressed as \(P(x) = a(x + 4)(x + 1)(x - 1)(x - 4)\), where \(a\) is a constant to be determined.

Step 3: Expand the Factored Form

First, multiply the pairs of conjugate terms: \((x + 4)(x - 4) = x^2 - 16\) and \((x + 1)(x - 1) = x^2 - 1\). Then combine: \(P(x) = a(x^2 - 16)(x^2 - 1)\).Multiply the two expressions: \[ P(x) = a(x^2 - 16)(x^2 - 1) = a(x^4 - x^2 - 16x^2 + 16) = a(x^4 - 17x^2 + 16) \].

Step 4: Use Additional Information to Find 'a'

We know \((-2, 10)\) is a point on the polynomial, meaning when \(x = -2\), \(P(-2) = 10\). Substitute into the polynomial to find \(a\): \[ 10 = a((-2)^4 - 17(-2)^2 + 16) \].Calculate the expression: \((-2)^4 = 16,\) and \((-2)^2 = 4\), so \(P(-2) = a(16 - 68 + 16) = a(-36)\).Thus, \(10 = -36a\), giving \(a = -\frac{5}{18}\).

Step 5: Write the Final Polynomial Function

Substitute the value of \(a\) back into the polynomial: \[ P(x) = -\frac{5}{18}(x^4 - 17x^2 + 16) \].

Key Concepts

Real Roots

Real roots of a polynomial function are the values of the variable that make the polynomial equal to zero. In simple terms, they are the points where the graph of the polynomial crosses or touches the x-axis. For instance, if you have a polynomial function, and you want to find its roots, you will set the function equal to zero and solve for the variable. This gives you the real roots. In the provided exercise, the roots given are

• \(-4\)
• \(-1\)
• \(1\)
• \(4\)

These roots suggest that the polynomial has four points where it intercepts the x-axis. Each real root corresponds to a factor of the polynomial. For example, a root of \(-4\) provides a factor of \((x + 4)\). Thus, knowing the real roots is crucial as they directly help us form the polynomial in its factored form.

Factored Form

Factored form is a way of expressing a polynomial as a product of its factors. It's like turning a big number into smaller, multiplication terms. For our polynomial with the roots \( -4, -1, 1, \) and \( 4, \) the factored form helps us see the underlying structure. Each root tells us one of these factors. So, we could write it as

• \( (x + 4)(x + 1)(x - 1)(x - 4) \)

Here, each binomial expression inside the parentheses represents a root of the polynomial. But, remember, since we don't have the full polynomial, we place a constant \(a\) in front, which we will determine later. Factored form is great because it immediately shows us the roots. It is especially useful for graphing and understanding the behavior of the polynomial function.

Constant Determination

Finding the constant \(a\) in the polynomial helps us determine the exact shape of the polynomial's graph. To do this, we use additional information, often another point that lies on the graph. In this case, it's \( (-2, 10) \). Plug \( x = -2 \) into the polynomial and set it equal to 10, because that's the y-value when \( x = -2 \). You substitute and solve for \(a\).
Here's how it works:

• Substitute \(x = -2\) into the polynomial formula we found.
• Calculate: \((-2)^4 - 17(-2)^2 + 16\) to find it equals \(-36\).
• Since \(10 = -36a\), find \(a = -\frac{5}{18}\).

This calculation aligns the polynomial with the specific coordinate point outside the roots, allowing the full polynomial expression to accurately reflect the given data.

Expanding Polynomials

Expanding polynomials involves multiplying out their factors to write them in standard polynomial form. This can be done by applying distributive properties and expanding each term accordingly. For instance, if we have our polynomial in factored form as

• \( a(x + 4)(x + 1)(x - 1)(x - 4) \)

First, multiply pairs of factors: \( (x + 4)(x - 4) = x^2 - 16\) and \( (x + 1)(x - 1) = x^2 - 1\). Now, multiply these results to get:

• \( a(x^2 - 16)(x^2 - 1) = a(x^4 - 17x^2 + 16) \)

This process of expanding helps clear out the parentheses and lays out each term separately. It reveals any combining needed between like terms and gives the polynomial its full expanded algebraic identity. Expanding is crucial to a horizontal view of the polynomial, simplifying analysis, calculations, and the derivation of further properties.