Problem 16 - Algebra and Trigonometry

Question

13- 30 . Factor the polynomial completely and find all its zeros. State the multiplicity of each zero. $$ Q(x)=x^{2}-8 x+17 $$

Step-by-Step Solution

Answer

The zeros are \( 4+i \) and \( 4-i \) with multiplicity 1.

Step 1: Check for Quadratic Form

Identify that the polynomial \( Q(x) = x^2 - 8x + 17 \) is a quadratic in the form \( ax^2 + bx + c \). In this case, \( a = 1 \), \( b = -8 \), and \( c = 17 \).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c \) is calculated using the formula \( \Delta = b^2 - 4ac \). Substitute the values: \( \Delta = (-8)^2 - 4(1)(17) = 64 - 68 = -4 \).

Step 3: Analyze the Discriminant

The discriminant \( \Delta = -4 \) is less than zero, indicating that the polynomial has no real roots and cannot be factored using real numbers.

Step 4: Find Complex Roots

Since the discriminant is negative, the roots are complex. Use the quadratic formula \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \). Here, \( x = \frac{-(-8) \pm \sqrt{-4}}{2(1)} = \frac{8 \pm 2i}{2} \).

Step 5: Simplify Complex Roots

Simplify the expression: \( x = \frac{8}{2} \pm \frac{2i}{2} = 4 \pm i \). Thus, the zeros are \( 4 + i \) and \( 4 - i \), both with multiplicity 1.

Key Concepts

Complex Roots

When we solve quadratic polynomials, especially those whose discriminant is negative, we often encounter complex roots. A complex number is a number that has both a real and an imaginary part, which can be expressed in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part.

In our exercise, the quadratic polynomial \( Q(x) = x^2 - 8x + 17 \) resulted in complex roots. This is because the discriminant was negative, which typically occurs when the parabola represented by the quadratic expression does not intersect the x-axis. The presence of complex roots signifies that the solutions exist in the set of complex numbers rather than the set of real numbers.

• Complex roots always appear in conjugate pairs for polynomials with real coefficients. In this case, the complex roots are \( 4 + i \) and \( 4 - i \).
• The imaginary unit \( i \) is defined as \( \sqrt{-1} \).
• Both complex roots have a multiplicity of 1, indicating that each appears only once as a solution of the polynomial.

Factoring Polynomials

Factoring polynomials involves expressing the polynomial as a product of its factors. However, not all polynomials can be factored using real numbers alone, particularly when they possess complex roots. In such cases, recognizing the polynomial's complex roots is crucial since they indicate that factoring can involve complex conjugates.

In the case of \( Q(x) = x^2 - 8x + 17 \), factoring over real numbers is impossible because it does not cross or touch the x-axis, evidenced by the negative discriminant. However, over the complex number set, it can be represented in terms of its roots:

• The complex roots \( 4 + i \) and \( 4 - i \) give us a factored form: \( (x - (4 + i))(x - (4 - i)) \).
• When expanded, these factors will translate back into the original quadratic polynomial.

Understanding when and how to factor polynomials is vital in solving equations since it allows for revealing possible solutions and characteristics of the graph, such as intercepts and turning points.

Discriminant Analysis

The discriminant is a powerful tool in determining the nature of roots for quadratic polynomials. Calculated using the formula \( \Delta = b^2 - 4ac \) for a polynomial \( ax^2 + bx + c \), it offers critical insights into the type of roots present.

In the exercise, the discriminant for \( Q(x) = x^2 - 8x + 17 \) was computed as \( -4 \), which is less than zero. This negative value is essential because:

• A negative discriminant indicates the absence of real roots and the presence of complex roots.
• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root, also known as a repeated root.

Being familiar with discriminant analysis aids in predicting the behaviour of polynomials graphically and algebraically, allowing more efficient problem-solving techniques concerning polynomial equations.