Math Problem Statement

Form a polynomial whose zeros and degree are given. Zeros: 6, multiplicity 1; -4, multiplicity 2; degree 3. Type a polynomial with integer coefficients and a leading coefficient of 1.

Solution

To form a polynomial with the given properties:

Given Information

  • Zeros:
    • 66 with multiplicity 11
    • 4-4 with multiplicity 22
  • Degree: 33
  • Leading coefficient: 11

Solution

  1. Write each factor for the given zeros and their multiplicities:

    • For zero 66 with multiplicity 11: the factor is (x6)(x - 6).
    • For zero 4-4 with multiplicity 22: the factor is (x+4)2(x + 4)^2.
  2. Combine the factors: f(x)=(x6)(x+4)2f(x) = (x - 6)(x + 4)^2

  3. Expand the expression: First, expand (x+4)2(x + 4)^2: (x+4)2=x2+8x+16(x + 4)^2 = x^2 + 8x + 16 Now multiply by (x6)(x - 6): f(x)=(x6)(x2+8x+16)f(x) = (x - 6)(x^2 + 8x + 16)

  4. **Distribute (x6)(x - 6) across each term in (x2+8x+16)(x^2 + 8x + 16): f(x)=x(x2+8x+16)6(x2+8x+16)f(x) = x(x^2 + 8x + 16) - 6(x^2 + 8x + 16) f(x)=x3+8x2+16x6x248x96f(x) = x^3 + 8x^2 + 16x - 6x^2 - 48x - 96

  5. Combine like terms: f(x)=x3+2x232x96f(x) = x^3 + 2x^2 - 32x - 96

Final Answer

The polynomial is: f(x)=x3+2x232x96f(x) = x^3 + 2x^2 - 32x - 96

This polynomial has integer coefficients, a leading coefficient of 11, and the correct zeros with specified multiplicities.

Would you like a deeper breakdown of any step?


Here are five related questions to deepen your understanding:

  1. What is the impact of a zero's multiplicity on the shape of a polynomial's graph?
  2. How would the polynomial change if the zero 4-4 had multiplicity 11 instead of 22?
  3. How can you confirm the degree of the polynomial after expansion?
  4. How does the leading coefficient affect the end behavior of the polynomial?
  5. What method can you use to factor polynomials when only given some of the roots?

Tip: When expanding polynomials, keep track of each term carefully to avoid errors in sign or coefficient.

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Math Problem Analysis

Mathematical Concepts

Polynomials
Roots and Zeros
Multiplicities

Formulas

f(x) = (x - a)^m * (x - b)^n

Theorems

Fundamental Theorem of Algebra

Suitable Grade Level

Grades 10-12