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Chapter 3: Problem 96

Why is it not necessary to apply the rational zero theorem, Descartes' rule of signs, or the upper and lower bound theorem to find the zeros of a second- degree polynomial?

Short Answer

Expert verified
The quadratic formula directly provides the zeros of a second-degree polynomial, making other methods unnecessary.

Step by step solution

01

- Understanding the Quadratic Formula

For any second-degree polynomial (quadratic function), the zeros can be found using the quadratic formula: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \] where the polynomial is given by \( ax^2 + bx + c = 0 \).
02

- Nature of Quadratic Polynomials

Quadratic polynomials always have at most two real zeros. This is because the highest power of the polynomial is 2, limiting the number of real roots to two.
03

- Direct Formula Application

The quadratic formula provides a direct method to determine the zeros without needing to use methods like the rational zero theorem, Descartes' rule of signs, or the upper and lower bound theorem. These techniques are useful for higher-degree polynomials where finding zeros is less straightforward.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Formula
The quadratic formula is a powerful tool to find the zeros of any second-degree polynomial. For any quadratic equation of the form \( ax^2 + bx + c = 0 \), the zeros can be determined using: \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \).This formula is derived from completing the square and provides a straightforward method.
  • \(a\), \(b\), and \(c\) are constants
  • \(\pm\) means there can be two solutions (one with \( + \), one with \( - \))
Using this formula, you can simply substitute the values of \(a\), \(b\), and \(c\), and easily calculate the zeros of the quadratic equation.
Real Zeros
Real zeros are the values of \(x\) that make the polynomial equal to zero. For a quadratic equation, the quadratic formula can produce either:
  • Two real and distinct zeros if \(b^2 - 4ac > 0\)
  • One real (repeated) zero if \(b^2 - 4ac = 0\)
  • No real zeros if \(b^2 - 4ac < 0\)
This discriminant \(b^2 - 4ac\) decides the nature of the zeros. If the discriminant is positive, it indicates two real solutions. If zero, it indicates a single real solution, and if negative, there are no real solutions.
Polynomial Functions
Polynomial functions are mathematical expressions involving sums of powers of \(x\) with coefficients. Quadratic functions are a type of polynomial function where the highest power of the variable \(x\) is 2. They can always be written in the form of \(ax^2 + bx + c\).
  • \(a\), \(b\), and \(c\) are coefficients
  • The highest power (2) determines that it is a quadratic polynomial
This specific structure of quadratic polynomials ensures that the method to find their zeros is significantly simpler compared to higher-degree polynomials.
Higher-Degree Polynomials
Higher-degree polynomials have terms with variable powers greater than 2, like cubics (degree 3) or quartics (degree 4). Finding the zeros of these polynomials is more complex because they can have more than two real zeros.
  • For cubic polynomials, the highest power of \(x\) is 3
  • For quartic polynomials, the highest power of \(x\) is 4
Methods like the Rational Zero Theorem, Descartes' Rule of Signs, and the Upper and Lower Bound Theorem are useful for these higher-degree polynomials. These techniques help to narrow down potential zeros by testing possible rational zeros, determining the number of positive and negative real zeros, and identifying boundaries for the zeros.

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Most popular questions from this chapter

Determine if the statement is true or false. If \(b\) is an upper bound for the real zeros of a polynomial, then \(-b\) is a lower bound for the real zeros of the polynomial.

a. Factor the polynomial over the set of real numbers. b. Factor the polynomial over the set of complex numbers. $$f(x)=x^{4}+2 x^{2}-35$$

The light from a lightbulb radiates outward in all directions. a. Consider the interior of an imaginary sphere on which the light shines. The surface area of the sphere is directly proportional to the square of the radius. If the surface area of a sphere with a \(10-\mathrm{m}\) radius is \(400 \pi \mathrm{m}^{2}\), determine the surface area of a sphere with a \(20-\mathrm{m}\) radius. b. Explain how the surface area changed when the radius of the sphere increased from \(10 \mathrm{~m}\) to \(20 \mathrm{~m}\). c. Based on your answer from part (b) how would you expect the intensity of light to change from a point \(10 \mathrm{~m}\) from the lightbulb to a point \(20 \mathrm{~m}\) from the lightbulb? d. The intensity of light from a light source varies inversely as the square of the distance from the source. If the intensity of a lightbulb is 200 lumen/m \(^{2}\) (lux) at a distance of \(10 \mathrm{~m}\), determine the intensity at \(20 \mathrm{~m}\).

Determine if the statement is true or false. If \(c\) is a zero of a polynomial \(f(x)\), with degree \(n \geq 2\) then all other zeros of \(f(x)\) are zeros of \(\frac{f(x)}{x-c}\).

The front face of a tent is triangular and the height of the triangle is two- thirds of the base. The length of the tent is \(3 \mathrm{ft}\) more than the base of the triangular face. If the tent holds a volume of \(108 \mathrm{ft}^{3}\), determine its dimensions.
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