Problem 15 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

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

Step-by-Step Solution

Answer

\(p_1(x)\) 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 1: 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: \(a=1\), \(b=-2\), \(c=1\).

Step 2: 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 \(p_3(x)\), \(\Delta = (-2)^2 - 4(1)(1) = 4 - 4 = 0\).

Step 3: 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.

Step 4: 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 \(b=-2\), \(a=1\): \(x = \frac{2}{2} = 1\).

Key Concepts

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.