Question

unit 2 modeling polynomial functions name practice 1 topic 1 possible number of real zeros and end behavior tell how many possible real zeros each function could have and describe the end behavior of each function. 1) $f(x)=x^3 - 2x^2 + 2$ 2) $f(x)=2x^2 - 12x + 14$ 3) $f(x)=x^5 - 2x^3 + x + 2$ 4) $f(x)=-x^4 + 4x^2 + 3x - 3$ 5) $f(x)=x^3 - x^2 + 1$ 6) $f(x)=-x^3 + x^2$ 7) $f(x)=-x^5 + 4x^3 - 5x - 2$ 8) $f(x)=-x^5 + 3x^3 - 2x + 1$ 9) $f(x)=-2x^2 + 8x - 8$ 10) $f(x)=x^4 - 3x^2 + 3x$

Answer:

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

Step 1: Possible Real Zeros (Descartes' Rule of Signs)

• For \( f(x) \): Number of sign changes in \( f(x) \): \( +x^3 - 2x^2 + 2 \) → 2 sign changes. So possible positive real zeros: 2 or 0.
• For \( f(-x) = -x^3 - 2x^2 + 2 \): Number of sign changes: 1. So possible negative real zeros: 1.
• Degree is 3, so total possible real zeros: 3 (2 positive + 1 negative, or 0 positive + 1 negative + 2 complex, but we count possible real: 1 or 3? Wait, Descartes' Rule: positive zeros: 2 or 0; negative: 1. So possible real zeros: 1 (0 positive + 1 negative) or 3 (2 positive + 1 negative). Wait, degree 3, so total real zeros: 1 or 3 (since complex zeros come in pairs). Wait, correction: positive zeros: 2 or 0; negative: 1. So possible real zeros: 1 (0 + 1) or 3 (2 + 1). But let's check end behavior.

Step 2: End Behavior

• Leading term: \( x^3 \), leading coefficient positive.
• As \( x \to \infty \), \( f(x) \to \infty \); as \( x \to -\infty \), \( f(x) \to -\infty \).

Problem 2: \( f(x) = 2x^2 - 12x + 14 \)

Step 1: Possible Real Zeros (Discriminant or Descartes' Rule)

• Descartes' Rule: \( f(x) \): 2 sign changes ( \( +2x^2 -12x +14 \) ), so positive zeros: 2 or 0. \( f(-x) = 2x^2 + 12x +14 \), 0 sign changes, so negative zeros: 0.
• Discriminant: \( D = (-12)^2 - 4(2)(14) = 144 - 112 = 32 > 0 \), so 2 real zeros.

Step 2: End Behavior

• Leading term: \( 2x^2 \), leading coefficient positive.
• As \( x \to \pm\infty \), \( f(x) \to \infty \).

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

Step 1: Possible Real Zeros (Descartes' Rule)

• \( f(x) \): \( +x^5 -2x^3 +x +2 \) → 2 sign changes ( \( + \) to \( - \), \( - \) to \( + \) ), so positive zeros: 2 or 0.
• \( f(-x) = -x^5 + 2x^3 -x + 2 \): sign changes: \( - \) to \( + \), \( + \) to \( - \), \( - \) to \( + \) → 3 sign changes, so negative zeros: 3 or 1.
• Degree 5, so possible real zeros: 1, 3, 5 (but considering positive: 2/0, negative: 3/1). So possible real zeros: 1 (0 + 1), 3 (2 + 1 or 0 + 3), 5 (2 + 3).

Step 2: End Behavior

• Leading term: \( x^5 \), leading coefficient positive.
• As \( x \to \infty \), \( f(x) \to \infty \); as \( x \to -\infty \), \( f(x) \to -\infty \).

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

Step 1: Possible Real Zeros (Descartes' Rule)

• \( f(x) \): \( -x^4 + 4x^2 + 3x - 3 \): sign changes: \( - \) to \( + \), \( + \) to \( - \) → 2 sign changes. Positive zeros: 2 or 0.
• \( f(-x) = -x^4 + 4x^2 - 3x - 3 \): sign changes: \( - \) to \( + \), \( + \) to \( - \) → 2 sign changes. Negative zeros: 2 or 0.
• Degree 4, so possible real zeros: 0, 2, 4.

Step 2: End Behavior

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

Problem 5: \( f(x) = x^3 - x^2 + 1 \)

Step 1: Possible Real Zeros (Descartes' Rule)

• \( f(x) \): \( +x^3 -x^2 +1 \): 2 sign changes. Positive zeros: 2 or 0.
• \( f(-x) = -x^3 -x^2 +1 \): 1 sign change. Negative zeros: 1.
• Degree 3, so possible real zeros: 1 (0 + 1) or 3 (2 + 1).

Step 2: End Behavior

• Leading term: \( x^3 \), leading coefficient positive.
• As \( x \to \infty \), \( f(x) \to \infty \); as \( x \to -\infty \), \( f(x) \to -\infty \).

Problem 6: \( f(x) = -x^3 + x^2 \)

Step 1: Possible Real Zeros (Factor)

• Factor: \( x^2(-x + 1) = -x^2(x - 1) \). So zeros at \( x = 0 \) (multiplicity 2) and \( x = 1 \). So real zeros: 2 (but \( x=0 \) is a double root). Wait, Descartes' Rule:
• \( f(x) \): \( -x^3 + x^2 \): 1 sign change. Positive zeros: 1.
• \( f(-x) = x^3 + x^2 \): 0 sign changes. Negative zeros: 0.
• So possible real zeros: 1 (but we see two real zeros, one with multiplicity 2). So possible real zeros: 1 or 2 (since multiplicity 2 is still a real zero).

Step 2: End Behavior

• Leading term: \( -x^3 \), leading coefficient negative.
• As \( x \to \infty \), \( f(x) \to -\infty \); as \( x \to -\infty \), \( f(x) \to \infty \).

Problem 7: \( f(x) = -x^5 + 4x^3 - 5x - 2 \)

Step 1: Possible Real Zeros (Descartes' Rule)

• \( f(x) \): \( -x^5 + 4x^3 - 5x - 2 \): sign changes: \( - \) to \( + \), \( + \) to \( - \) → 2 sign changes. Positive zeros: 2 or 0.
• \( f(-x) = x^5 - 4x^3 + 5x - 2 \): sign changes: \( + \) to \( - \), \( - \) to \( + \), \( + \) to \( - \) → 3 sign changes. Negative zeros: 3 or 1.
• Degree 5, so possible real zeros: 1 (0 + 1), 3 (2 + 1 or 0 + 3), 5 (2 + 3).

Step 2: End Behavior

• Leading term: \( -x^5 \), leading coefficient negative.
• As \( x \to \infty \), \( f(x) \to -\infty \); as \( x \to -\infty \), \( f(x) \to \infty \).

Problem 8: \( f(x) = -x^5 + 3x^3 - 2x + 1 \)

Step 1: Possible Real Zeros (Descartes' Rule)

• \( f(x) \): \( -x^5 + 3x^3 - 2x + 1 \): sign changes: \( - \) to \( + \), \( + \) to \( - \), \( - \) to \( + \) → 3 sign changes. Positive zeros: 3 or 1.
• \( f(-x) = x^5 - 3x^3 + 2x + 1 \): sign changes: \( + \) to \( - \), \( - \) to \( + \) → 2 sign changes. Negative zeros: 2 or 0.
• Degree 5, so possible real zeros: 1 (3 + 0 - 2? Wait, no: positive: 3 or 1; negative: 2 or 0. So possible real zeros: 1 (1 + 0), 3 (3 + 0), 3 (1 + 2), 5 (3 + 2). Wait, degree 5, so total real zeros: 1, 3, or 5.

Step 2: End Behavior

• Leading term: \( -x^5 \), leading coefficient negative.
• As \( x \to \infty \), \( f(x) \to -\infty \); as \( x \to -\infty \), \( f(x) \to \infty \).

Problem 9: \( f(x) = -2x^2 + 8x - 8 \)

Step 1: Possible Real Zeros (Factor or Discriminant)

• Factor: \( -2(x^2 - 4x + 4) = -2(x - 2)^2 \). So zero at \( x = 2 \) (multiplicity 2). So real zero: 1 (with multiplicity 2).
• Descartes' Rule: \( f(x) \): \( -2x^2 + 8x - 8 \): 2 sign changes. Positive zeros: 2 or 0. But we see a double root, so 1 real zero (with multiplicity 2).

Step 2: End Behavior

• Leading term: \( -2x^2 \), leading coefficient negative.
• As \( x \to \pm\infty \), \( f(x) \to -\infty \).

Problem 10: \( f(x) = x^4 - 3x^2 + 3x \)

Step 1: Possible Real Zeros (Factor)

• Factor: \( x(x^3 - 3x + 3) \). So \( x = 0 \) is a zero. For \( x^3 - 3x + 3 \), use Descartes' Rule:
• \( f(x) = x^3 - 3x + 3 \): 2 sign changes. Positive zeros: 2 or 0.
• \( f(-x) = -x^3 + 3x + 3 \): 1 sign change. Negative zeros: 1.
• So possible real zeros: 1 (from \( x=0 \)) + 1 (negative) + 0 or 2 (positive) → total 2, 3, or 4? Wait, degree 4, so possible real zeros: 1, 2, 3, or 4? Wait, \( x=0 \) is one zero. Then \( x^3 - 3x + 3 \) has degree 3, so possible real zeros: 1 or 3. So total real zeros: 1 + 1 = 2, 1 + 3 = 4, or 0 + 1 = 1 (but \( x=0 \) is a zero, so at least 1).

Step 2: End Behavior

• Leading term: \( x^4 \), leading coefficient positive.
• As \( x \to \pm\infty \), \( f(x) \to \infty \).

Summary of Answers (for each problem, possible real zeros and end behavior):

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

• Possible real zeros: 1 or 3
• End behavior: \( x \to \infty \), \( f(x) \to \infty \); \( x \to -\infty \), \( f(x) \to -\infty \)

1. \( f(x) = 2x^2 - 12x + 14 \)

• Possible real zeros: 2 (discriminant \( 32 > 0 \))
• End behavior: \( x \to \pm\infty \), \( f(x) \to \infty \)

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

• Possible real zeros: 1, 3, or 5
• End behavior: \( x \to \infty \), \( f(x) \to \infty \); \( x \to -\infty \), \( f(x) \to -\infty \)

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

• Possible real zeros: 0, 2, or 4
• End behavior: \( x \to \pm\infty \), \( f(x) \to -\infty \)

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

• Possible real zeros: 1 or 3
• End behavior: \( x \to \infty \), \( f(x) \to \infty \); \( x \to -\infty \), \( f(x) \to -\infty \)

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

• Possible real zeros: 2 ( \( x=0 \) (multiplicity 2) and \( x=1 \) )
• End behavior: \( x \to \infty \), \( f(x) \to -\infty \); \( x \to -\infty \), \( f(x) \to \infty \)

1. \( f(x) = -x^5 + 4x^3 - 5x - 2 \)

• Possible real zeros: 1, 3, or 5
• End behavior: \( x \to \infty \), \( f(x) \to -\infty \); \( x \to -\infty \), \( f(x) \to \infty \)

1. \( f(x) = -x^5 + 3x^3 - 2x + 1 \)

• Possible real zeros: 1, 3, or 5
• End behavior: \( x \to \infty \), \( f(x) \to -\infty \); \( x \to -\infty \), \( f(x) \to \infty \)

1. \( f(x) = -2x^2 + 8x - 8 \)

• Possible real zeros: 1 ( \( x=2 \), multiplicity 2 )
• End behavior: \( x \to \pm\infty \), \( f(x) \to -\infty \)

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

• Possible real zeros: 1, 2, 3, or 4 (at least 1 from \( x=0 \))
• End behavior: \( x \to \pm\infty \), \( f(x) \to \infty \)