Unit 5 Polynomial Functions Homework 1 Answer Key

Unit 5 Polynomial Functions Homework 1 Answer Key

Polynomial functions form a cornerstone of algebra and pre‑calculus curricula, and Unit 5 often focuses on mastering their graphs, zeros, end‑behavior, and factoring techniques. Homework 1 in this unit typically introduces students to the foundational skills needed to manipulate and interpret polynomials before moving on to more complex applications such as polynomial division, the Remainder and Factor Theorems, and real‑world modeling. Having a reliable answer key not only lets learners verify their work but also illuminates the logical steps that lead to each solution, turning a simple check‑mark into a deeper learning opportunity.

Overview of Unit 5 Polynomial Functions

Before diving into the specific problems, it helps to recall the key concepts that Homework 1 usually targets:

Homework 1 usually presents a mix of straightforward identification tasks (degree, leading coefficient) and basic factoring problems that reinforce these ideas.

Typical Homework 1 Problems and Their Solutions

Below is a representative set of questions that often appear in Unit 5, Polynomial Functions – Homework 1, followed by a detailed answer key. The numbering mirrors common textbook layouts, but the methods apply universally.

Problem 1 – Identify Degree and Leading Coefficient

1a. (P(x)= -5x^4 + 3x^2 - 7)
1b. (Q(x)= 2x^7 - x^5 + 4x - 9)

Answer Key

• 1a. Degree = 4 (largest exponent). Leading coefficient = -5 (coefficient of (x^4)).
• 1b. Degree = 7. Leading coefficient = 2.

Why it matters: Knowing the degree and leading coefficient lets you predict end‑behavior without graphing.

Problem 2 – Find the Zeros by Factoring

2. (R(x)= x^3 - 6x^2 + 11x - 6)

Answer Key

1. Look for rational roots using the Rational Root Theorem: possible roots are (\pm1, \pm2, \pm3, \pm6).

2. Test (x=1): (1-6+11-6=0) → (x=1) is a zero.

3. Perform synthetic division of (R(x)) by ((x-1)):

   [ \begin{array}{r|rrrr} 1 & 1 & -6 & 11 & -6 \ & & 1 & -5 & 6 \ \hline & 1 & -5 & 6 & 0 \end{array} ]

   Quotient: (x^2 -5x +6).

4. Factor the quadratic: (x^2-5x+6 = (x-2)(x-3)).

Zeros: (x=1,; x=2,; x=3) (each of multiplicity 1).

Problem 3 – Determine End‑Behavior

3. Describe the end‑behavior of (S(x)= -3x^5 + 4x^3 - x + 2).

Answer Key - Degree = 5 (odd).

• Leading coefficient = -3 (negative).

For odd degree with a negative leading coefficient:

• As (x\to -\infty), (S(x)\to +\infty) (left end rises).
• As (x\to +\infty), (S(x)\to -\infty) (right end falls).

Problem 4 – Multiplicity and Graph Behavior

4. Given (T(x)= (x+2)^2 (x-4)^3), state the zeros, their multiplicities, and whether the graph crosses or touches the x‑axis at each zero.

Answer Key

Problem 5 – Write a Polynomial from Given Zeros

5. Write a polynomial of least degree with real coefficients that has zeros at (-1) (multiplicity 2) and (3) (multiplicity 1).

Answer Key

• Start with factors: ((x+1)^2) for the double zero, ((x-3)) for the single zero.

• Multiply:

  [ P(x)= (x+1)^2 (x-3) = (x^2+2x+1)(x-3) ]

• Expand (optional):

  [ P(x)= x^3 - x^2 -5x -3 ]

Any non‑zero constant multiple of this polynomial is also correct; the simplest form uses leading coefficient = 1.

Problem 6 – Apply the Remainder Theorem

6. Find the remainder when (U(x)= 2x^4 - 3x^3 + x^2 - 5x + 7) is divided by ((x-2)).

Answer Key

The Remainder Theorem states that the remainder of (U(x)) divided by ((x-a)) equals (U(a)).

• Compute (U

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Problem 6 – Apply the Remainder Theorem (Continued)

6. Find the remainder when (U(x)= 2x^4 - 3x^3 + x^2 - 5x + 7) is divided by ((x-2)).

Answer Key

The Remainder Theorem states that the remainder of (U(x)) divided by ((x-a)) equals (U(a)).

• Compute (U(2)):
  (U(2) = 2(2)^4 - 3(2)^3 + (2)^2 - 5(2) + 7 = 2(16) - 3(8) + 4 - 10 + 7 = 32 - 24 + 4 - 10 + 7 = 11)

• Therefore, the remainder is 11.

Problem 7 – Synthetic Division

7. Use synthetic division to divide (V(x)= 4x^3 - 5x^2 + 2x - 1) by ((x+1)).

Answer Key

1. Set up the synthetic division with the divisor (-1) and the coefficients of V(x):

   -1 |  4  -5   2  -1
      |      -4   9  -11
      ------------------
         4  -9  11   0
   

2. The result is (Q(x) = 4x^2 - 9x + 11) with a remainder of 0.

Problem 8 – Graphing Polynomials

8. Sketch a graph of (W(x) = (x-1)(x+2)^3).

Answer Key (This would typically include a graph, but we’ll describe the key features)

• Zeros: x = 1 (multiplicity 1, crosses the x-axis) and x = -2 (multiplicity 3, touches and turns at the x-axis).
• End Behavior: As x → -∞, W(x) → -∞. As x → +∞, W(x) → +∞.
• Turning Points: The graph will have two turning points (local max and local min) due to the multiplicity of the factors.

Conclusion

This exploration of polynomial functions has covered fundamental concepts, including identifying degree and leading coefficients, finding zeros, determining end behavior, and applying key theorems like the Remainder Theorem and Synthetic Division. We’ve seen how these tools allow us to analyze and predict the characteristics of polynomial graphs, and how the zeros and multiplicities of a polynomial directly influence its behavior. Mastering these techniques is crucial for understanding a wide range of applications in mathematics, science, and engineering. Further study can delve into more complex polynomial types, such as rational, irreducible, and reciprocal polynomials, as well as techniques for solving polynomial equations and analyzing their roots in greater detail. Remember, a solid grasp of polynomial functions forms a cornerstone for more advanced mathematical concepts.

Problem 9 – The Factor Theorem

9. Use the Factor Theorem to determine whether ((x-3)) is a factor of (P(x)= x^3 - 4x^2 + x + 6).

Answer Key

The Factor Theorem states that ((x-a)) is a factor of (P(x)) if and only if (P(a)=0).

• Compute (P(3)):
  (P(3) = (3)^3 - 4(3)^2 +