Notes on Polynomial and Rational Functions

Learning Objectives

  • Evaluate a polynomial using the Remainder Theorem.

  • Use the Factor Theorem to solve a polynomial equation.

  • Use the Rational Zero Theorem to find rational zeros.

  • Find zeros of a polynomial function.

  • Use Linear Factorization Theorem to find polynomials with given zeros.

  • Use Descartes' Rule of Signs.

  • Solve real-world applications of polynomial equations.

Zeros of Polynomial Functions

  • Example Problem: Determine the dimensions of a cake with a volume of 351 cubic inches, where the length is four inches longer than the width and the height is one-third of the width.

  • Approach: Write a cubic function representing the volume of the cake:

    • Let width = w, then length = w + 4, height = w/3.

    • Volume equation: V = w(w + 4)(w/3) = 351.

    • Expand and rearrange to form a polynomial equation, solve for w using various methods discussed.

Evaluating a Polynomial Using the Remainder Theorem

  • Remainder Theorem Statement: If a polynomial f(x) is divided by x − k, the remainder is f(k).

  • Steps for Evaluation:

    1. Use synthetic division to divide f(x) by x − k.

    2. The remainder equals f(k).

  • Example: For f(x) = 6x^4 − x^3 − 15x^2 + 2x − 7, evaluating at x = 2:

    • Synthetic Division:

    1. Divide: Calculate values step by step to ensure clarity.

    2. Result: The final step yields a remainder of 25.

    • Conclusion: Thus, f(2) = 25 represents the evaluation at that point.

Using the Factor Theorem to Solve Polynomial Equations

  • Factor Theorem Statement: k is a zero of f(x) if and only if (x − k) is a factor of f(x).

  • Approach:

    1. Perform synthetic division by (x − k).

    2. If the remainder is 0, (x − k) is a confirmed factor.

    3. Factor further if applicable.

  • Example: Confirm (x + 2) as a factor of x^3 − 6x^2 − x + 30:

    • Division: Conduct synthetic division and simplify step by step, confirming factors.

    • Conclusion: Resulting factors: (x + 2)(x - 3)(x - 5). Thus, the zeros found are -2, 3, and 5.

Using the Rational Zero Theorem to Find Rational Zeros

  • Rational Zero Theorem Statement: If f(x) has integer coefficients, every rational zero has the form p/q, where p is a factor of the constant term a0 and q is a factor of the leading coefficient an.

  • Procedure:

    1. Identify factors of the constant term and leading coefficient:

    • a0 = -4 (factors: ±1, ±2, ±4)

    • an = 2 (factors: ±1, ±2)

    1. Construct potential rational zeros: ±1, ±2, ±4, ±1/2.

    2. Test each potential zero using synthetic division and discuss outcomes.

  • Example: For f(x) = 2x^4 − 5x^3 + x^2 − 4, possible rational zeros include ±1, ±2, ±4, ±1/2. Substitute these values in and evaluate to find usable solutions.

Finding the Zeros of Polynomial Functions

  • To find zeros:

    1. Apply the Rational Zero Theorem to list possible rational zeros.

    2. Use synthetic division to test these candidates one by one.

    3. Continue factoring the resulting polynomial till reaching a quadratic, applying the quadratic formula if necessary.

  • Example: With f(x) = 4x^3 − 3x − 1, apply the Rational Zero Theorem, test each value via synthetic division, and discover real roots like 1 and -1/2 with their respective multiplicities.

Using the Fundamental Theorem of Algebra

  • Statement: Every polynomial function of degree n has exactly n complex roots (counting multiplicities).

  • Implication: Break down polynomial into linear factors corresponding to the zeros found.

  • Example: For f(x) = 3x^3 + 9x^2 + x + 3, identify rational zeros using previous methods; then explore any complex solutions if necessary.

Using the Linear Factorization Theorem

  • Linear Factorization Theorem: A polynomial function will contain as many factors as its degree with the form (x − c).

  • Conjugate Pairs: Any complex zero of the form a + bi will necessarily have its conjugate a - bi also as a zero, which keeps coefficients real in such circumstances.

Descartes’ Rule of Signs

  • Application: Use this rule to estimate the number of positive and negative real zeros by counting sign changes in f(x) and f(-x).

  • Example: For f(x) = −x^4 − 3x^3 + 6x^2 − 4x − 12, noting two sign changes indicates two or zero positive real zeros; conduct evaluation of additional values accordingly for comprehensive understanding.

Solving Real-World Applications

  • Strategy: Translate real-world problems into polynomial equations using relationships provided.

  • Example: Bakery problem where dimensions lead to cubic equations:

    Develop equations based on context and solve to find necessary dimensions, concluding with calculated values of length = 13 inches, width = 9 inches, height = 3 inches which satisfy the volume condition.