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Magazine Chapter 2: Problem 21
If \(x\) is real, then the expression \(\frac{x^{2}+34 x-71}{x^{2}+2 x-7}\) (A) cannot lie between 5 and 9 (B) always lies between 5 and 9 (C) is not real (D) None of these
Short Answer
Expert verified
(A) The expression cannot strictly lie between 5 and 9 due to real intersection points.
Step by step solution
01
Analyze Polynomial Denominator
The expression is given as \(\frac{x^{2}+34x-71}{x^{2}+2x-7}\). First, note that the expression is undefined when the denominator is zero. Let's find the roots of the denominator, \(x^{2}+2x-7=0\). Using the quadratic formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\), with \(a=1, b=2, c=-7\), we find the roots.
02
Solve the Quadratic Equation for Denominator
Calculate the discriminant: \(2^2 - 4 \times 1 \times (-7) = 4 + 28 = 32\). This is positive, indicating two real roots. The roots are \(x = \frac{{-2 \pm \sqrt{32}}}{2} \). Simplifying \(\sqrt{32} = 4\sqrt{2}\), we have \(x = -1 \pm 2\sqrt{2}\). Thus, the expression is undefined at \(x = -1 + 2\sqrt{2}\) and \(x = -1 - 2\sqrt{2}\).
03
Determine Asymptotic and Critical Values
Now consider the behavior of the expression as \(x\) approaches infinity. In this case, the dominant terms, \(x^2\), cancel out, and the expression approaches \(1\). As \(x\) approaches the roots \(-1 + 2\sqrt{2}\) and \(-1 - 2\sqrt{2}\), the expression tends towards \(\pm\infty\), due to the denominator approaching 0.
04
Evaluate the Range of the Expression
The expression can be rewritten as \(1 + \frac{32x - 78}{x^2 + 2x - 7}\). Analyze this to see where it could equal \(5\) and \(9\).\(5\) and \(9\) will correspond to the equation: \(\frac{32x - 78}{x^2 + 2x - 7} = k - 1\), where \(k=5\) or \(9\). This must be solved for allowable \(x\) to see if \(5 < \text{expression} < 9\) is possible.
05
Solve Critical Points for Expression Sytematically
Set \(1 + \frac{32x - 78}{x^2 + 2x - 7} = 5\) and \(= 9\). Solve the resulting equations to find \(x\). The transformations yield quadratics: for \(5\): \(32x - 78 = 4(x^2 + 2x - 7)\), similar steps for 9. Solving these: \(x^2 -16x +50=0\) (for lower), and \(x^2 -20x+50=0\) (for upper). Their discriminants \(16^2-4*50=196\) and \(20^2-4*50=300\) confirmed solutions in real x-values.
06
Check If Possible
The roots of these quadrics yield real x-intersections with \(x\) for 5 and beyond implies that certain \(5<x<9\) values are possible due to behavior beyond critical roots. Given expression cannot lie strictly 5 to 9 without real intersections, it intersects certain numbers beyond thresholds.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Division
Polynomial division is a technique used to simplify expressions where one polynomial is divided by another. When you encounter an expression like \( \frac{x^{2}+34x-71}{x^{2}+2x-7} \), it involves polynomials in both the numerator and the denominator.
- You begin by understanding the degree of each polynomial 鈥� here, both are quadratic (degree 2).
- The process resembles long division with numbers, but it's with variables involved. You divide the leading term of the numerator by the leading term of the denominator.
- The division continues until you can't divide anymore, often leaving a remainder.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). If factoring seems complex or not straightforward, the quadratic formula saves the day.
To solve for \( x \), use:
\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]
To solve for \( x \), use:
\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]
- The term \( b^2 - 4ac \) is known as the discriminant, and it determines the nature of the roots.
- If the discriminant is positive, there are two real and distinct roots.
- If zero, there's one real root (a repeated root), and if negative, the roots are complex.
- In our case, for the denominator \( x^2 + 2x - 7 \), the discriminant was positive, leading to two real solutions \( x = -1 \pm 2\sqrt{2} \).
Undefined Expressions
When working with rational expressions like \( \frac{x^{2}+34x-71}{x^{2}+2x-7} \), it's crucial to consider where they might become undefined, a common issue in math.
- An expression becomes undefined when the denominator equals zero since division by zero is mathematically invalid.
- Using the quadratic formula, it was found that the denominator \( x^2+2x-7 \) equals zero when \( x = -1 \pm 2\sqrt{2} \), giving us the points where the expression is undefined.
- At these points, the graph of the function would have a vertical asymptote, meaning the value of the expression approaches either infinity or negative infinity as \( x \) approaches these values.
