Graphing Polynomials

Problem: Let polynomial \( P \) be given by \( P(x) = -x^3 - x^2 + 2x \).

1. Factor polynomial \( P \).
2. Determine the multiplicity of each zero of \( P \).
3. Determine the sign chart of \( P \).
4. Graph polynomial \( P \) and label the \( x \)- and \( y \)-intercepts.

Solution:

1. Factor \( P \) as follows: \[ P(x) = -x^3 - x^2 + 2x = -x(x^2 + x - 2) = -x(x + 2)(x - 1) \]
2. \( P \) has three zeros which are \( -2 \), \( 0 \), and \( 1 \), all with multiplicity one.
3. The three zeros of \( P \) split the number line into four intervals: \[ (-\infty, -2), \quad (-2, 0), \quad (0, 1), \quad (1, +\infty) \] Select test values in each interval to determine the sign: \[ P(-3) = 12 \gt 0, \quad P(-1) = -2 \lt 0, \quad P\left(\frac{1}{2}\right) = \frac{5}{8} \gt 0, \quad P(2) = -8 \lt 0 \]

   

   

4. Using the sign chart and intercepts, sketch the graph:

   

Problem: Let polynomial \( P \) be given by \( P(x) = x^4 - 2x^2 + 1 \).

1. Factor polynomial \( P \).
2. What is the multiplicity of each zero of \( P \)?
3. Determine the sign chart of \( P \).
4. Graph polynomial \( P \) and label the \( x \)- and \( y \)-intercepts.
5. What is the range of polynomial \( P \)?

Solution:

1. Factor \( P \) as follows: \[ P(x) = x^4 - 2x^2 + 1 = (x^2 - 1)^2 = ((x - 1)(x + 1))^2 = (x - 1)^2(x + 1)^2 \]
2. Polynomial \( P \) has zeros at \( x = 1 \) and \( x = -1 \), both with multiplicity 2.
3. Polynomial \( P(x) \) is a perfect square and therefore nonnegative for all real numbers \( x \). It equals zero at \( -1 \) and \( 1 \), and is positive everywhere else:

   

4. The \( x \)-intercepts are at \( (-1, 0) \) and \( (1, 0) \), and the \( y \)-intercept is at \( (0, 1) \). The graph touches the \( x \)-axis at these points and opens upward:

   

5. From the graph, the range of \( P \) is given by the interval: \[ [0, +\infty) \]

Problem: Let polynomial \( P \) be given by \( P(x) = x^4 + 5x^3 + 5x^2 - 5x - 6 \).

1. Show that \( x = -3 \) is a zero of \( P \).
2. Show that \( (x - 1) \) is a factor of \( P \).
3. Factor \( P \) completely and determine the multiplicity of each zero.
4. Determine the sign chart of \( P \).
5. Graph polynomial \( P \) and label the intercepts.

Solution:

1. Calculate \( P(-3) \): \[ P(-3) = (-3)^4 + 5(-3)^3 + 5(-3)^2 - 5(-3) - 6 = 81 - 135 + 45 + 15 - 6 = 0 \] Hence, \(-3\) is a zero, and \( x + 3 \) is a factor of \( P(x) \).
2. Dividing \( P(x) \) by \( x + 3 \) gives quotient \( x^3 + 2x^2 - x - 2 \). Dividing that quotient by \( x - 1 \) yields \( x^2 + 3x + 2 \) with a remainder of 0, proving \( x - 1 \) is also a factor.
3. Writing \( P(x) \) in fully factored form: \[ P(x) = (x + 3)(x - 1)(x^2 + 3x + 2) = (x + 3)(x - 1)(x + 1)(x + 2) \] \( P \) has zeros at \( x = -3, -2, -1, \) and \( 1 \), all of multiplicity one.
4. Sign chart:

   

5. Graphing using zeros and sign chart:

   

Problem: Given that \( x = 1 \) is a zero of multiplicity 2 of \( P(x) = x^5 + x^4 - 3x^3 - x^2 + 2x \), construct a sign chart and graph it.

Solution:

Dividing \( P(x) \) by \( (x - 1)^2 \) yields quotient \( x^3 + 3x^2 + 2x \). Factoring completely: \[ P(x) = (x - 1)^2(x^3 + 3x^2 + 2x) = x(x - 1)^2(x^2 + 3x + 2) = x(x - 1)^2(x + 1)(x + 2) \]

• \( P(x) \) has 4 zeros at \( x = -2, -1, 0, \) and \( 1 \), with \( x = 1 \) having multiplicity 2.
• Sign chart:

  

• Graph: