Polynomial and Rational Functions Flashcards

Power and Polynomial Functions\n\nA power function is defined by the form $f(x) = x^{p}$. Toolkit power functions include the constant ($f(x) = x^{0}$ ), identity ($f(x) = x^{1}$), quadratic, and cubic functions. Even whole number powers ($n = 2, 4, 6, \dots$) possess graphs that flatten near the origin and are symmetric, with long run behavior described as $f(x) \to \infty$ for $x \to \pm\infty$. Odd whole number powers ($n = 3, 5, 7, \dots$) approach $\infty$ as $x \to \infty$ and $-\infty$ as $x \to -\infty$. A polynomial is a sum of transformed power functions with non-negative whole number powers: $f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + \dots + a_{1}x + a_{0}$. The degree of a polynomial is the highest power $n$, and the leading term determines its long run behavior. Polynomials of degree $n$ have at most $n$ horizontal intercepts and $n-1$ turning points.\n\n# Quadratic Functions\n\nQuadratic functions are second-degree polynomials. They are commonly represented in standard form, $f(x) = ax^{2} + bx + c$, or transformation/vertex form, $f(x) = a(x - h)^{2} + k$. The vertex $(h, k)$ represents the maximum or minimum point of the parabola and can be found algebraically using $h = -\frac{b}{2a}$ and $k = f(h)$. The horizontal intercepts are identified by solving $f(x) = 0$ via factoring or the quadratic formula: $x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$. The coefficient $a$ determines the direction of the opening: if a > 0, the vertex is a minimum, and if a < 0, the vertex is a maximum.\n\n# Short Run Behavior and Division Theorems\n\nShort run behavior involves the analysis of horizontal and vertical intercepts. Horizontal intercepts (zeros or roots) relate to the factors of the polynomial. The multiplicity of a zero, determined by the power $p$ on a factor $(x - h)^{p}$, dictates the graph's behavior at the intercept: multiplicity 1 passes through, multiplicity 2 bounces, and multiplicity 3 flattens while passing through the axis. Polynomial division, including long division and synthetic division, is used to factor higher-degree polynomials. The Remainder Theorem states that the remainder of $p(x) \div (x - c)$ is equal to $p(c)$. The Factor Theorem specifies that a real number $c$ is a zero of $p(x)$ if and only if $(x - c)$ is a factor of the polynomial.\n\n# Real and Complex Zeros\n\nCauchy's Bound determines an interval $[-(\frac{M}{|a_{n}|} + 1), \frac{M}{|a_{n}|} + 1]$ where all real zeros of a polynomial exist, with $M$ being the largest absolute coefficient. The Rational Roots Theorem identifies potential rational roots as $\pm \frac{p}{q}$, where $p$ are factors of the constant and $q$ are factors of the leading coefficient. While some polynomials lack real zeros, the Fundamental Theorem of Algebra guarantees that any degree $n$ polynomial has at least one real or complex zero. Counting multiplicities, a polynomial of degree $n$ has exactly $n$ zeros. Complex numbers are expressed as $z = a + bi$, where $i = \sqrt{-1}$. For polynomials with real coefficients, non-real zeros always occur in complex conjugate pairs: $a + bi$ and $a - bi$.\n\n# Rational Functions\n\nRational functions are ratios of two polynomials expressed as $f(x) = \frac{P(x)}{Q(x)}$. Vertical asymptotes occur where the denominator equals zero, provided the numerator is non-zero at that point. If both the numerator and denominator are zero, the graph may exhibit a hole. Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator: if the denominator degree is higher, the asymptote is $y = 0$; if the degrees are equal, the asymptote is the ratio of the leading coefficients. If the numerator's degree is exactly one higher than the denominator's, an oblique asymptote (or slant asymptote) exists and is found using the quotient from polynomial division.\n\n# Inverses and Radical Functions\n\nRadical functions arise from the inversion of power and polynomial functions. Finding the inverse of a quadratic function requires restricting its domain—typically to $x \ge h$—to ensure it is one-to-one. The resulting inverse is a square root function. To find the inverse of a rational function, variables are swapped and the resulting equation is solved algebraically for the output. Radical functions are often defined by fractional exponents (e.g., $x^{\frac{1}{2}}$). When finding the domain of radical or rational expressions, one must ensure that the values under even roots are non-negative and denominators are non-zero.","title":"Chapter 3: Polynomial and Rational Functions"}

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