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Chapter 4: Problem 15

Wieviele Nullstellen (und gegebenenfalls welche) besitzen die Polynome \(p_{1}(x)=x^{2}+x+1\), \(p_{2}(x)=2 x^{2}+6 x-20, p_{3}(x)=x^{2}-2 x+1 ?\)

Short Answer

Expert verified
\(p_1(x)\) has no real roots; \(p_2(x)\) has roots at \(x=2\) and \(x=-5\); \(p_3(x)\) has a root at \(x=1\).

Step by step solution

01

Identify Coefficients for Each Quadratic Polynomial

For a quadratic polynomial, the standard form is \(ax^2 + bx + c\). In \(p_1(x) = x^2 + x + 1\), coefficients are: \(a=1\), \(b=1\), \(c=1\). In \(p_2(x) = 2x^2 + 6x - 20\), coefficients are: \(a=2\), \(b=6\), \(c=-20\). In \(p_3(x) = x^2 - 2x + 1\), coefficients are: \(a=1\), \(b=-2\), \(c=1\).
02

Determine the Discriminant for Each Polynomial

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is calculated as \(\Delta = b^2 - 4ac\). For \(p_1(x)\), \(\Delta = 1^2 - 4(1)(1) = 1 - 4 = -3\). For \(p_2(x)\), \(\Delta = 6^2 - 4(2)(-20) = 36 + 160 = 196\). For \(p_3(x)\), \(\Delta = (-2)^2 - 4(1)(1) = 4 - 4 = 0\).
03

Evaluate the Number of Solutions Using the Discriminant

The discriminant provides information about the number and nature of the roots. If \(\Delta > 0\), there are two distinct real roots. If \(\Delta = 0\), there is one real root (a repeated root). If \(\Delta < 0\), there are no real roots, but two complex roots. For \(p_1(x)\), \(\Delta = -3\), so no real solutions. For \(p_2(x)\), \(\Delta = 196\), so two distinct real solutions. For \(p_3(x)\), \(\Delta = 0\), so one real solution.
04

Solve for the Real Roots (if any)

For \(p_2(x)\), use the quadratic formula: \[x = \frac{-b \pm \sqrt{\Delta}}{2a}\]. Substitute \(b=6\), \(\Delta=196\), \(a=2\): \[x = \frac{-6 \pm \sqrt{196}}{4} = \frac{-6 \pm 14}{4}\]. The roots are \(x = 2\) and \(x = -5\). For \(p_3(x)\), since \(\Delta = 0\), the root is given by \(x = \frac{-b}{2a}\). Substitute \(b=-2\), \(a=1\): \(x = \frac{2}{2} = 1\).

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Key Concepts

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

Discriminant
The discriminant is a key component in understanding quadratic equations. It helps determine the nature and the number of roots a quadratic equation will have. The formula for the discriminant, denoted as \( \Delta \), is \( \Delta = b^2 - 4ac \).
  • If \( \Delta > 0 \), the quadratic equation has two distinct real roots. This indicates that the graph of the equation touches the x-axis at two separate points.
  • If \( \Delta = 0 \), there is exactly one real root, often called a double or repeated root. This means the graph is tangent to the x-axis at a single point.
  • If \( \Delta < 0 \), there are no real roots; instead, the equation has two complex conjugate roots. Here, the graph does not intersect the x-axis at all.
For example, in the polynomial \( p_1(x) = x^2 + x + 1 \), the discriminant is \( -3 \). Because it is negative, there are no real roots, only complex ones.
Polynomial Roots
In mathematics, the roots of a polynomial are the values of \( x \) that make the polynomial equal to zero. For a quadratic polynomial, these can be found using various methods like factoring, completing the square, or the quadratic formula.
  • In equations like \( p_2(x) = 2x^2 + 6x - 20 \), finding roots involves calculating where the graph of the polynomial crosses the x-axis.
  • Once the roots become known, they provide solutions to the equation when set to zero.
In this exercise, we found that \( p_2(x) = 2x^2 + 6x - 20 \) has roots \( x = 2 \) and \( x = -5 \). These are found where the polynomial evaluates to zero, representing the two points where it crosses the x-axis.
Real and Complex Roots
Understanding the nature of roots is crucial in solving quadratic equations. Real roots are those that can be plotted on a number line, while complex roots involve imaginary numbers and cannot be directly visualized on the standard number line.
  • Real roots occur when the discriminant is non-negative (\( \Delta \geq 0 \)). They may occur as two distinct points or as one repeated root.
  • Complex roots come in pairs and occur when the discriminant is negative (\( \Delta < 0 \)), making use of the imaginary unit \( i \) where \( i^2 = -1 \).
For example, in \( p_1(x) = x^2 + x + 1 \), due to a negative discriminant, the roots are complex. Hence, they are not visible as points of intersection with the x-axis on a real number line.
Quadratic Formula
The quadratic formula is a universal method for solving any quadratic equation \( ax^2 + bx + c = 0 \). The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
  • This formula provides a straightforward way to find the roots without needing to factor the polynomial.
  • It incorporates the discriminant within the square root, making it pivotal in determining the type of roots (real or complex).
  • For \( p_2(x) \), using the quadratic formula yielded roots \( x = 2 \) and \( x = -5 \).
With the quadratic formula, one can always determine the roots of a quadratic equation, provided the coefficients are known, ensuring that every quadratic can be addressed efficiently.

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Most popular questions from this chapter

{Sei} p(z) ;}=\sum_{k=O} a_{k} z^{K}\( ein komplexes Polynom mit reellen Koeffizienten (d.h. \)a_{k} \in \mathbb{R}\( f眉r \)\left.k=0,1, \ldots, n\right)\(. Zeigen Sie: Ist \)z_{0}\( eine Nullstelle von p, so ist auch \)\bar{z}_{0}\( eine Nullstelle von \)p .$

skizzieren sie die Graphen der Funktion \(g t[-2,4] \rightarrow \mathbb{R}\) sit \(g(x) ;=(x-1)^{2}-4\) und von \(|g|=\)

Berechnen sie mit Hilfe des Horner-Schemas die Funktionswerte \(p(2)\) und \(p(-2)\) f眉r das polynom \(p(x)=x^{4}-5 x^{3}+5 x^{2}+5 x-6\)

Welche der folgenden Funktionen ist beschr盲nkt? Geben Sie nur im Fall a) eine mathematische Begr眉ndung Ihrer Aussage. a) \(x \mapsto f(x):=\frac{1}{x}, x \geq 2\); b) \(x \mapsto g(x):=\sqrt{r^{2}-x^{2}},|x| \leq r, r>0\) c) \(x \longmapsto h(x):=x^{2}-3, x \in \mathbb{R}\).
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