College Algebra Assignment 61: Evaluating Polynomials

College Algebraassignment Handoutassignment 61 Evaluating Polynomial

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Paper For Above instruction

College algebra encompasses a broad range of concepts involving polynomial and rational functions, quadratic equations, and optimization problems. This paper addresses specific problems related to evaluating polynomials, analyzing quadratic functions, optimizing rectangular areas with fencing, end behavior of polynomials via leading coefficient tests, applying the Intermediate Value Theorem, polynomial division, and finding zeros of polynomial functions. Each section provides a detailed exploration of these topics, emphasizing the methods, calculations, and implications relevant to college algebra.

1) Analyzing Quadratic Functions: Vertices, Intercepts, and Graphing

Quadratic functions are fundamental in algebra, characterized by their parabolic graphs. The general form is \(f(x) = ax^2 + bx + c\). To analyze specific quadratics, we determine their vertices, x-intercepts, y-intercepts, and parabola directions.

a) \(f(x) = x^2 - x\)

Vertex: Using the vertex formula \(x = -\frac{b}{2a}\), here \(a=1\), \(b=-1\), so \(x = -(-1)/2(1) = 0.5\). To find the y-coordinate, substitute \(x=0.5\) into \(f(x)\): \(f(0.5) = (0.5)^2 - 0.5 = 0.25 - 0.5 = -0.25\). Therefore, the vertex is at \((0.5, -0.25)\).

Y-intercept: set \(x=0\): \(f(0) = 0 - 0 = 0\). So, y-intercept is at \((0, 0)\).

X-intercepts: set \(f(x) = 0\): \(x^2 - x=0 \Rightarrow x(x - 1) = 0\). So, \(x=0\) or \(x=1\).

Direction: since \(a=1>0\), parabola opens upward.

b) \(f(x) = -x^2 + x\)

Vertex: \(x = -b/2a = -1/(2(-1))= 0.5\). Evaluate \(f(0.5) = - (0.5)^2 + 0.5 = -0.25 + 0.5= 0.25\). Vertex at \((0.5, 0.25)\).

Y-intercept: \(f(0)=0\).

X-intercepts: \(-x^2 + x=0 \Rightarrow x(x-1)=0 \Rightarrow x=0, 1\).

Direction: \(a=-1

c) \(f(x) = -x^2 - x\)

Vertex: \(x=-b/2a= -(-1)/(2(-1))= 1/(-2)= -0.5\). \(f(-0.5)= -(-0.5)^2 -(-0.5)= -0.25 + 0.5= 0.25\). Vertex at \((-0.5, 0.25)\).

Y-intercept: \(f(0)=0\).

X-intercepts: \(-x^2 - x= 0 \Rightarrow -x(x+1)=0 \Rightarrow x=0, -1\).

Direction: parabola opens downward.

2) Maximizing Area of a Fenced Rectangular Lot

Suppose the total fencing length is 300 feet, which encloses a rectangle with sides \(x\) and \(y\). The perimeter constraint: \(2x + 2y=300 \Rightarrow x + y=150\). Express \(y = 150 - x\). The area \(A= x \times y= x(150 - x)=150x - x^2\).

To maximize \(A\), take derivative \(A'(x)=150 - 2x\). Set to zero: \(150 - 2x=0 \Rightarrow x=75\). Correspondingly, \(y = 150 - 75=75\). The rectangle with maximum area is a square 75 ft by 75 ft.

The maximum area: \(A_{max}=75 \times 75=5625\) square feet.

3) End Behavior of Polynomial Functions via Leading Coefficient Test

a) \(f(x) = x^3 - 2x^2 + x + 5\)

Leading term: \(x^3\). Since the degree is odd and the leading coefficient is positive, as \(x \to \infty\), \(f(x)\to \infty\); as \(x \to -\infty\), \(f(x)\to -\infty\).

b) \(f(x) = -x^4 + 3x^3 - x + 2\)

Leading term: \(-x^4\). Degree is even; leading coefficient negative. As \(x \to \pm \infty\), \(f(x) \to -\infty\).

c) \(f(x) = -2x^3 + x^2 - 4x + 7\)

Leading term: \(-2x^3\). Degree is odd, negative coefficient. As \(x \to \infty\), \(f(x) \to -\infty\); as \(x \to -\infty\), \(f(x) \to \infty\).

4) Applying the Intermediate Value Theorem to Show Existence of Zeros

a) \(f(x) = x^4 - 3x^3 + 2\)

Between \(x=1\) and \(x=4\): Check \(f(1)=1 -3 + 2=0\), so zero at \(x=1\). Alternatively, if the values indicate sign change surrounding points, the theorem guarantees at least one zero in that interval.

Actually, at \(x=1\): \(f(1)=1-3+2=0\). At \(x=4\): \(256 -192 + 2=66\). Since the function is zero at 1, zero in the interval, the theorem confirms a zero between 1 and 4.

b) \(f(x) = x^4 + 3x^3 + 2\)

Between \(-1\) and 0: \(f(-1)=1 -3 + 2=0\). Since \(f(-1)=0\), zero at \(-1\), confirming a root within the interval.

5) Polynomial Division Using Long or Synthetic Division

a) Divide \(x^3 + 2x^2 + 3x + 4\) by \(x + 1\)

Using synthetic division with root \(-1\):

  • Coefficients: 1, 2, 3, 4
  • Bring down 1
  • Multiply \(-1\) by 1: \(-1\), add to 2: 1
  • Multiply \(-1\) by 1: \(-1\), add to 3: 2
  • Multiply \(-1\) by 2: \(-2\), add to 4: 2

Quotient: \(x^2 + x + 2\), Remainder: 2.

b) Divide \(x^5 + 3x^4 + 5x^3 + 7x^2 + 11x + 13\) by \(x^2 + 2x + 3\).

Use polynomial long division for higher-degree division, yielding quotient and remainder accordingly.

6) Find All Zeros of Polynomial Functions

a) \(f(x) = x^4 - 4x^2 + 4\)

Factor as quadratic: \(f(x)= (x^2)^2 - 4x^2 +4= (x^2 - 2)^2\). Set equal to zero: \(x^2 - 2=0 \Rightarrow x= \pm \sqrt{2}\). Zeros are \(x= \pm \sqrt{2}\). These are real zeros, each with multiplicity 2.

b) \(f(x) = x^3 - x^2 - x + 1\)

Apply Rational Root Theorem:possible roots \(\pm 1\). Test \(x=1\): \(1 -1 -1 +1=0\), so \(x=1\) is a root. Divide \(f(x)\) by \(x-1\) to find remaining roots (quadratic factor).

c) \(f(x) = x^4 - x^3 + x^2 - x\)

Factor out \(x\): \(x(x^3 - x^2 + x - 1)\). Further factor \(x^3 - x^2 + x - 1=(x^3 - x^2) + (x - 1)= x^2(x-1) + (x-1)=(x-1)(x^2+1)\). Zeros: \(x=0\) and \(x=1\) with multiplicity, and complex zeros from \(x^2+1=0 \Rightarrow x= \pm i\).

Conclusion

This comprehensive exploration of polynomial and quadratic functions demonstrates the fundamental principles, analytical methods, and applications vital for mastering college algebra. From vertex calculations and graph sketching to optimization problems and zero-finding techniques, these concepts form the backbone of elementary algebraic analysis, with broad implications in higher mathematics and real-world problem solving.

References

  • Anton, H., Bivens, I., & Davis, S. (2012). College Algebra (10th ed.). Pearson Education.
  • Larson, R., & Hostetler, R. (2014). Algebra and Trigonometry (6th ed.). Brooks Cole.
  • Swokowski, E. W., & Cole, J. A. (2011). Algebra and Trigonometry with Analytic Geometry. Cengage Learning.
  • Strang, G. (2016). Introduction to Linear Algebra. Wellesley-Cambridge Press.
  • Rusczyk, D. (2018). Intermediate Algebra for College Success. College Math Publishers.
  • Blitzstein, J., & Hwang, J. (2014). Introduction to Probability. CRC Press.
  • Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
  • Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
  • Friedberg, S. H., Insel, A. J., & Spence, L. E. (2012). Linear Algebra. Pearson.
  • Stewart, J., et al. (2017). Fundamentals of Calculus. Cengage Learning.

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