Polynomial Functions: Factor Theorem, Rational Zeros, and Conjugate Zeros Theorem

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: $f(-3) = 6(-3)^3 + 19(-3)^2 + 2(-3) - 3 = -162 + 171 - 6 - 3 = 0$ 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. $f(x) = (x - 1)(x^2 - 2x - 6)$ Further factoring: $x^2 - 2x - 6 = (x - 3)(x + 2)$ So, $f(x) = (x - 1)(x - 3)(x + 2)$

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 $(x - 5i)(x + 5i) = x^2 + 25$ Set $x^4 + 16x^2 - 225 = 0$ Let $y = x^2$, so $y^2 + 16y - 225 = 0$ Solve for $y$ using the quadratic formula: $y = \frac{-16 \pm \sqrt{16^2 + 4 \times 225}}{2}$ $y = \frac{-16 \pm \sqrt{256 + 900}}{2}$ $y = \frac{-16 \pm 32}{2}$ $y = 8$ or $y = -24$ So, $x^2 = 8 \Rightarrow x = \pm 2\sqrt{2}$ $x^2 = -24 \Rightarrow x = \pm 2\sqrt{6}i$ Zeros: $x = 5i, -5i, 2\sqrt{2}, -2\sqrt{2}$

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: $\frac{\text{factors of constant term}}{\text{factors of leading coefficient}}$

• Example: For f(x) = 6x^3 + 7x^2 - 12x^2 - 3x + 2: Constant term = 2, factors: $\pm1, \pm2$ Leading coefficient = 6, factors: $\pm1, \pm2, \pm3, \pm6$ Possible rational zeros: $\pm1, \pm2, \pm\frac{1}{2}, \pm\frac{1}{3}, \pm\frac{2}{3}, \pm\frac{1}{6}, \pm\frac{2}{6}$

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: $\pm1, \pm2, \pm4, \pm8$ Test $x = -1$: $f(-1) = (-1)^3 - 5(-1)^2 + 2(-1) + 8 = -1 - 5 + (-2) + 8 = 0$ So, $x + 1$ is a factor. Divide to get $x^2 - 6x + 8$ Factor $x^2 - 6x + 8 = (x - 2)(x - 4)$ So, $f(x) = (x + 1)(x - 2)(x - 4)$

Summary Table: Polynomial Theorems and Methods

Theorem/Method	Statement	Application
Factor Theorem	If $f(k) = 0$, then $x - k$ is a factor of $f(x)$	Testing if a value is a zero; factoring polynomials
Conjugate Zeros Theorem	If $a + bi$ is a zero and coefficients are real, $a - bi$ is also a zero	Finding all zeros of polynomials with complex roots
Rational Zeros Theorem	Possible rational zeros are $\pm\frac{\text{factors of constant term}}{\text{factors of leading coefficient}}$	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.