Factoring and Graphing Polynomial Functions

Factoring Expressions for Graphing and Analysis

Introduction

The following notes outline the steps involved in factoring various polynomial expressions, determining their roots, and sketching graphs that include features such as roots, end behavior, and multiplicity.

Expressions to Factor

Expression 1: ( 2x^{2} + 11x + 21 )

• Factoring Process: Identify coefficients: ( a = 2, b = 11, c = 21 ).
• Use the quadratic formula or factoring techniques to find roots.

Expression 2: ( x^3 + 3x^2 + 30x + 90 )

• Factoring Process: Look for common factors and use polynomial long division or synthetic division for cubic expressions.

Expression 3: ( 25x^2 - 64 )

• Factoring Process: Recognize as a difference of squares:
  • Use the identity: ( a^2 - b^2 = (a - b)(a + b) )
  • Therefore: ( 25x^2 - 64 = (5x - 8)(5x + 8) )

Expression 4: ( x^2 + x + 2 )

• Factoring Process: Check for factorability. In this case, use the quadratic formula as it does not factor cleanly:
  • Roots: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )

Functions for Root Analysis and Graphing

Function 5: ( f(x) = x^3 + 9x )

• Factoring: Factor out the common term: ( f(x) = x(x^2 + 9) )
• Roots: Determine that roots exist at ( x = 0 ).
  • The quadratic does not yield real roots since ( (x^2 + 9) ) has no real solutions.
• End Behavior: As ( x \to \pm \, \infty ), ( f(x) \to \, + \infty ).
• Multiplicity of root at ( x = 0 ) is odd (1).

Function 6: ( f(x) = 2x^4 - 20x^2 + 18 )

• Factoring: Factor out the greatest common factor ( 2 ):
  • ( f(x) = 2(x^4 - 10x^2 + 9) )
  • Set ( u = x^2 ) for simplification:
  • Solve ( u^2 - 10u + 9 = 0 ) using the quadratic formula.
• Roots: Determine actual ( x ) values from ( u ).
• End Behavior: As ( x \to \pm \, \infty ), ( f(x) \to +
  \infty ).

Function 7: ( f(x) = x^2 - x - 42 )

• Factoring: Look for pairs of factors of ( -42 ) that sum to ( -1 ):
  • Roots identified as ( (x - 7)(x + 6) = 0 )
• Roots: ( x = 7 ) and ( x = -6 )
• End Behavior: As ( x \to \, -
  \infty ), ( f(x) \to +
  \infty ).

Function 8: ( f(x) = -2x^3 + 4x^2 + 16x )

• Factoring: Factor out common factor ( -2x ):
  • ( f(x) = -2x(x^2 - 2x - 8) )
• Roots: Find roots of the quadratic using the quadratic formula or factored form.
• End Behavior: As ( x \to \pm \, \infty ), function decreases to negative infinity and increases as well. This suggests a local maximum or minimum.

Function 9: ( f(x) = 2x^2 + 13x - 7 )

• Factoring: Identify coefficients and apply the quadratic formula:
  • ( x = \frac{-13 \pm \sqrt{13^2 - 4\cdot2\cdot(-7)}}{2\cdot2} )
• Roots: Solve for the exact roots.
• End Behavior: Determine the behavior at the extremes based on the leading coefficient.

Function 10: ( f(x) = x^3 + 3x^2 - 9x - 27 )

• Factoring: Use polynomial long division or synthetic division if possible.
• Roots: Find roots for this cubic function, determine behavior at the roots.
• End Behavior: As ( x \to \pm \, \infty ), consider the sign of leading term to determine behavior.

Summary of Graphing Results

• Determine roots accurately and check multiplicity for local minima and maxima.
• Analyze end behavior based on leading coefficients.
• Sketch graphs taking into account all the gathered information from factoring expressions, including roots and multiplicity.