Problem 66 - College Algebra and Calculus: An Applied Approach

Question

Modeling Polynomials A fourth-degree polynomial function \(g\) has real zeros \(-2,0,1\), and \(5 .\) Find two different polynomial functions, one with a positive leading coefficient and one with a negative leading coefficient, that could be \(g .\) How many different polynomial functions are possible for \(g\) ?

Step-by-Step Solution

Answer

The two different polynomial functions could be \(g(x) = (x + 2)x(x - 1)(x - 5)\) and \(g(x) = -(x + 2)x(x - 1)(x - 5)\). The amount of different polynomial functions possible for \(g\) is 512.

Step 1: Formulate Polynomial with Positive Leading Coefficient

Translate the zeros into factors of a polynomial by subtracting each zero from \(x\) and multiplicate them. The polynomial with a positive leading coefficient that corresponds to zeros \(-2,0,1,5\) is \(g(x) = (x+2)x(x-1)(x-5)\).

Step 2: Formulate Polynomial with Negative Leading Coefficient

For a polynomial with negative leading coefficient, simply put a negative sign in front of the polynomial from Step 1: \(g(x) = - (x+2)x(x-1)(x-5)\). This formula corresponds to the same zeros but the shape of the graph would be inverted compared to the first.

Step 3: Determine the Number of Different Polynomial Functions Possible

For a fourth degree polynomial, each root could have a multiplicity from 1 to 4. Therefore, without considering the leading coefficient, there are \(4^4 = 256\) different polynomial functions possible, considering multiplicity of roots. When considering the leading coefficient could be positive or negative, the number doubles to 512.

Key Concepts

Fourth-degree polynomial

Polynomial functions are mathematical expressions that involve variables raised to whole-number exponents. A fourth-degree polynomial is particularly interesting because its highest term is raised to the power of four. This gives it specific characteristics. A polynomial of degree four can be expressed as:\[ g(x) = ax^4 + bx^3 + cx^2 + dx + e \]where \(a, b, c, d,\) and \(e\) are constants, with \(a\) being nonzero. Given that the degree of this polynomial is four, it has up to four distinct solutions or roots. These roots decide where the polynomial crosses the x-axis.Fourth-degree polynomials can exhibit complex behavior. For instance, based on its coefficients, it might have complex roots or multiplicities. But, generally, when talking about real roots or zeros, the graph will cut or touch the x-axis accordingly based on these real zeros. The behavior of the graph also depends on the sign and magnitude of the leading coefficient.

Real zeros

Real zeros are the x-values where the polynomial equals zero. For a polynomial of degree four, there can be up to four real zeros, signifying the points where the graph of the polynomial intersects the x-axis. In the case presented, the real zeros are

• -2
• 0
• 1
• 5

Each zero represents a factor of the polynomial. To find the polynomial that correlates with these zeros, you would set up factors like \[ (x + 2), x, (x - 1), \,\text{and}\, (x - 5) \]Multiplying these factors together gives you a polynomial expression where these values of \(x\) make \(g(x)\) equal to zero.Real zeros can also be repeated, giving roots a multiplicity greater than one. This means that while you can have different polynomial functions with the same zeros, the exact structure of the polynomial may vary.

Leading coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree. It plays a crucial role in determining the end behavior of the polynomial's graph.For a fourth-degree polynomial like\[ g(x) = a(x+2)x(x-1)(x-5) \]Here, \(a\) is the leading coefficient. For instance, if \(a\) is positive, as in the polynomial \[ g(x) = (x+2)x(x-1)(x-5) \]the graph tends to rise on both ends due to its quartic nature. Conversely, if \(a\) is negative, as in\[ g(x) = - (x+2)x(x-1)(x-5) \]the graph will fall on both ends.The leading coefficient is also integral in determining the stretch and orientation of the graph. A larger magnitude means a steeper graph, while the sign determines if the endpoints will be in agreement (both up or both down) or in disagreement (one up, one down). Therefore, choosing the sign of \(a\) allows flexibility in the behavior of the polynomial's graph.