BackPolynomial Functions: Factor Theorem, Rational Zeros, and Conjugate Zeros Theorem
Study Guide - Smart Notes
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Polynomial Functions and Their Zeros
Factor Theorem
The Factor Theorem is a fundamental result in algebra that connects the zeros of a polynomial to its factors. It states that for any polynomial function f(x), x - k is a factor of the polynomial if and only if f(k) = 0.
Key Point: If substituting x = k into f(x) yields zero, then x - k is a factor of f(x).
Example: Is x + 3 a factor of f(x) = 6x^3 + 19x^2 + 2x - 3? Substitute x = -3: Since f(-3) = 0, x + 3 is a factor.
Factoring Polynomials Using the Factor Theorem
The Factor Theorem can be used to factor polynomials of higher degree into linear factors of the form x - b.
Key Point: If k is a zero of f(x), then x - k is a factor.
Example: Factor f(x) = x^3 - 3x^2 - 5x + 6 given that 1 is a zero. Synthetic division or long division can be used to factor out x - 1. Further factoring: So,
Conjugate Zeros Theorem
The Conjugate Zeros Theorem states that if a polynomial function f(x) has only real coefficients and if a + bi is a zero (where a and b are real numbers), then its conjugate a - bi is also a zero of f(x).
Key Point: Complex zeros of polynomials with real coefficients always occur in conjugate pairs.
Example: For p(x) = x^4 + 16x^2 - 225, given that 5i is a zero, find the remaining zeros. Since 5i is a zero, so is -5i. Factor out Set Let , so Solve for using the quadratic formula: or So, Zeros:
Potential Rational Zeros
The Rational Zeros Theorem provides a way to list all possible rational zeros of a polynomial with integer coefficients. The possible rational zeros are given by:
Formula:
Example: For f(x) = 6x^3 + 7x^2 - 12x^2 - 3x + 2: Constant term = 2, factors: Leading coefficient = 6, factors: Possible rational zeros:
Finding Rational Zeros and Factoring Polynomials
To find all rational zeros of a polynomial, use the Rational Zeros Theorem to list candidates, then test each by substitution or synthetic division. Once zeros are found, factor the polynomial into linear factors.
Example: For f(x) = x^3 - 5x^2 + 2x + 8: Possible rational zeros: Test : So, is a factor. Divide to get Factor So,
Summary Table: Polynomial Theorems and Methods
Theorem/Method | Statement | Application |
|---|---|---|
Factor Theorem | If , then is a factor of | Testing if a value is a zero; factoring polynomials |
Conjugate Zeros Theorem | If is a zero and coefficients are real, is also a zero | Finding all zeros of polynomials with complex roots |
Rational Zeros Theorem | Possible rational zeros are | Listing candidates for rational zeros |
Synthetic Division | Efficient method for dividing polynomials by linear factors | Testing zeros and factoring polynomials |
Additional info: Synthetic division steps and examples were expanded for clarity. The summary table was inferred to organize the main theorems and methods discussed in the notes.
