Problem 62 - Precalculus Mathematics for Calculus

Question

Find a polynomial of the specified degree that has the given zeros. Degree \(5 ; \quad\) zeros -2,-1,0,1,2

Step-by-Step Solution

Answer

The polynomial is \(x^5 - 5x^3 + 4x\).

Step 1: Identify all zeros

We are given the zeros of the polynomial: -2, -1, 0, 1, and 2. Since the polynomial degree is 5, and there are 5 zeros given, we can conclude that these are all the roots of the polynomial.

Step 2: Form the factors from zeros

Use each zero to form a linear factor of the polynomial. A polynomial zero \(c\) translates to the factor \( (x - c) \). Therefore, the factors will be: \( (x + 2), (x + 1), x, (x - 1), (x - 2) \).

Step 3: Form the polynomial

Multiply all the factors together to find the polynomial: \[(x + 2)(x + 1)x(x - 1)(x - 2)\] which expands to \[x(x^2 - 1)(x^2 - 4)\].

Step 4: Expand the polynomial

Expand the function \[x(x^2 - 1)(x^2 - 4)\] to verify its form:First, expand \((x^2 - 1)(x^2 - 4) = x^4 - 4x^2 - x^2 + 4 = x^4 - 5x^2 + 4\).Multiply by \(x\) to finalize the polynomial:\[x(x^4 - 5x^2 + 4) = x^5 - 5x^3 + 4x\].

Step 5: Write the final polynomial

The polynomial of degree 5 with the given zeros is \[x^5 - 5x^3 + 4x\].

Key Concepts

Degree of Polynomial

The degree of a polynomial is a fundamental concept in algebra. It tells us the highest power of the variable in a polynomial. In this exercise, we are dealing with a polynomial of degree 5. This means the highest exponent of the variable, in the polynomial, is 5. Therefore, the leading term will typically have the form \(ax^5\), where \(a\) is a coefficient.
This concept is crucial as it also indicates the number of possible roots or zeros the polynomial can have. Here, we were provided with the zeros: \(-2, -1, 0, 1,\) and \(2\). These correspond directly with the polynomial's degree of 5, confirming that all roots are given.

Zeros of Polynomial

Zeros, or roots, of a polynomial are the values of \(x\) for which the polynomial evaluates to zero. In simple terms, these are the x-values where the graph of the polynomial crosses the x-axis. In our problem, we're given the zeros as \(-2, -1, 0, 1,\) and \(2\).

• Each zero represents a solution to the polynomial equation \(p(x) = 0\).
• These zeros can also be used to determine the linear factors of the polynomial.
• For every zero \(c\), there is a corresponding factor \((x - c)\).

Knowing the zeros is incredibly helpful because they give us a direct way to factor the polynomial as explained in the following section.

Factors of a Polynomial

Factors are expressions that multiply together to form a polynomial. From the zeros given, \(-2, -1, 0, 1,\) and \(2\), we form the linear factors \((x + 2), (x + 1), x, (x - 1),\) and \((x - 2)\). Each factor represents a zero's corresponding point where the polynomial equals zero.

• For \(x = -2\), the factor is \((x + 2)\).
• For \(x = -1\), the factor is \((x + 1)\).
• The root \(x = 0\) corresponds to the factor \(x\), since \(0\) doesn’t change the sign.
• For \(x = 1\), the factor is \((x - 1)\).
• For \(x = 2\), the factor is \((x - 2)\).

Once all factors are established, you multiply them to create the full polynomial.

Expansion of Polynomials

The expansion of a polynomial involves multiplying its factors out to write it in its standard form. For the given problem, after finding the factors \((x + 2)(x + 1)x(x - 1)(x - 2)\), we need to expand them to arrive at a full polynomial.
Start by multiplying the simpler parts first, for example, \[x(x^2 - 1)(x^2 - 4)\]. Begin from the innermost brackets, and work your way out:

• Expand \((x^2 - 1)(x^2 - 4) = x^4 - 4x^2 - x^2 + 4\).
• This simplifies to \(x^4 - 5x^2 + 4\).
• Finally, multiply the result by \(x\) to achieve \(x(x^4 - 5x^2 + 4) = x^5 - 5x^3 + 4x\).

This step-by-step multiplication process not only verifies the polynomial’s form but also ensures accuracy in expanding all factors appropriately.