Problem 60 - Precalculus Mathematics for Calculus

Question

Find a polynomial of the specified degree that has the given zeros. Degree \(4 ; \quad\) zeros -2,0,2,4

Step-by-Step Solution

Answer

The polynomial is \(P(x) = x^4 - 4x^3 - 4x^2 + 16x\).

Step 1: Understand the Zeros

The given zeros of the polynomial are -2, 0, 2, and 4. These values are the roots of the polynomial equation. Since we are finding a polynomial of degree 4, each zero corresponds to a factor of the polynomial.

Step 2: Write the Polynomial in Factored Form

Using the zeros, write the polynomial in its factored form. Each zero \(-2, 0, 2, 4 \) corresponds to a factor: \(x+2, x, x-2, \) and \(x-4\). Thus, the polynomial can be expressed as:\[P(x) = (x + 2)(x)(x - 2)(x - 4)\]

Step 3: Multiply the Factors

Start multiplying the factors to find the polynomial in expanded form. Begin by multiplying the first two factors, followed by the last two, and then the two resulting expressions:\[(x+2)(x) = x^2 + 2x\]\[(x-2)(x-4) = x^2 - 4x - 2x + 8 = x^2 - 6x + 8\]

Step 4: Expand the Polynomial

Multiply the two quadratic expressions obtained in the previous step:\[(x^2 + 2x)(x^2 - 6x + 8) = x^4 - 6x^3 + 8x^2 + 2x^3 - 12x^2 + 16x\]Combine like terms to simplify:\[x^4 - 4x^3 - 4x^2 + 16x\]

Step 5: Write the Final Polynomial

The fully expanded polynomial is:\[P(x) = x^4 - 4x^3 - 4x^2 + 16x\] This is the polynomial of degree 4 with the given zeros.

Key Concepts

Factored Form

When a polynomial is expressed in a way that illustrates its roots (or zeros), it is said to be in its "factored form." This representation is beneficial because it shows exactly where the graph of the polynomial will intersect the x-axis.

• In the example given, the zeros are -2, 0, 2, and 4. Each of these zeros gives us a linear factor:
  • -2 corresponds to the factor \(x+2\)
  • 0 leads to the factor \(x\)
  • 2 gives the factor \(x-2\)
  • 4 results in the factor \(x-4\)

Formally speaking, if \( r \) is a root of the polynomial, then \((x - r)\) is a factor. Therefore, a polynomial of degree 4 with these zeros can be expressed as:\[ P(x) = (x+2)(x)(x-2)(x-4) \]This factored form is particularly insightful when considering the behavior of polynomials, as it makes influences such as multiplicity of roots clear.

Polynomial Expansion

Polynomial expansion involves taking a polynomial in its factored form and expanding it into a sum of terms. This process of expanding allows us to clearly see the degree and the leading coefficients which tell us a lot about the shape and orientation of the polynomial's graph.

• In the example solution, expansion was started by multiplying pairs of linear factors to convert the polynomial into quadratic expressions.
  • First, \( (x+2)(x) = x^2 + 2x \)
  • Then, \( (x-2)(x-4) = x^2 - 6x + 8 \)

The final step is to multiply these quadratic expressions to achieve a single polynomial:\[ (x^2 + 2x)(x^2 - 6x + 8) = x^4 - 6x^3 + 8x^2 + 2x^3 - 12x^2 + 16x \]Combining like terms leads to:\[ x^4 - 4x^3 - 4x^2 + 16x \]This is the polynomial in its expanded form, making it easier to analyze for calculus purposes such as derivative and integral calculations.

Zeros of Polynomial

The zeros of a polynomial are the values at which the polynomial evaluates to zero. In simpler terms, they are the x-values where the graph of the polynomial touches or crosses the x-axis.

• For the polynomial in question, the zeros given were -2, 0, 2, and 4.
  • These zeros translated directly into the factors \(x+2\), \(x\), \(x-2\), and \(x-4\).
  • If a graph of \(P(x)\) were drawn, it would touch or cross the x-axis at these points.

Zeros are fundamental in both algebra and calculus as they provide solutions to the polynomial equation \(P(x) = 0\). Understanding zeros offers insights into the polynomial's shape and structure, its potential maxima and minima, and even helps in integration calculations.