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Explain how factored form helps identify key features.
Easily identifies real zeros (x-intercepts), domain, vertical asymptotes, holes, and provides insight into the range.
How does standard form determine end behavior of polynomials?
Even degree with positive leading coefficient: ends go up. Even degree with negative leading coefficient: ends go down. Odd degree with positive leading coefficient: starts down, ends up. Odd degree with negative leading coefficient: starts up, ends down.
How does standard form determine end behavior of rational functions?
Compare degrees of numerator and denominator to find horizontal asymptotes, or the possibility of a slant asymptote.
Explain the purpose of polynomial long division.
To divide one polynomial by another, which helps in finding slant asymptotes and rewriting rational functions.
How does Pascal's Triangle relate to the Binomial Theorem?
Pascal's Triangle provides the coefficients for the terms in the binomial expansion.
Explain the significance of the remainder in polynomial long division.
The remainder helps in expressing the original rational function as a sum of a polynomial and a simpler rational function, useful for identifying asymptotes.
What does the graph of a rational function reveal?
Maxima, minima, points of inflection, shape, symmetry, asymptotes, and intercepts.
How do you identify holes in a rational function from its factored form?
Holes occur when a factor cancels out from both the numerator and denominator.
How do you find the slant asymptote of a rational function?
Perform polynomial long division. If the degree of the numerator is one greater than the degree of the denominator, the quotient is the equation of the slant asymptote.
What are the key features to analyze when graphing a rational function?
Zeros, y-intercept, vertical asymptotes, horizontal or slant asymptotes, and end behavior.
What is factored form of a polynomial?
Polynomial expressed as a product of its factors.
What is standard form of a polynomial?
Polynomial written in descending order of powers.
Define real zeros of a function.
The x-intercepts of the function's graph.
What are vertical asymptotes?
Vertical lines where the function approaches infinity or negative infinity.
What are removable discontinuities (holes)?
Points where a function is not defined, but the limit exists.
Define polynomial long division.
A method to divide one polynomial by another.
What is the quotient in polynomial division?
The result of dividing one polynomial by another, excluding the remainder.
What is the remainder in polynomial division?
The polynomial left over after polynomial long division, with a degree less than the divisor.
What is the Binomial Theorem?
A method to expand expressions of the form $(a + b)^n$.
Define binomial coefficient.
The coefficients in the expansion of $(a+b)^n$, often found using Pascal's Triangle.
What is the formula for the Binomial Theorem?
$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$
How to express polynomial division result?
$f(x) = g(x)q(x) + r(x)$ where $q(x)$ is the quotient and $r(x)$ is the remainder.
What is the form for finding slant asymptotes after polynomial division?
$f(x) = q(x) + \frac{r(x)}{g(x)}$, where $q(x)$ represents the slant asymptote if the degree of $g(x)$ is 1.
What is the condition for a horizontal asymptote at y=0?
Numerator degree < denominator degree.
What is the condition for a horizontal asymptote at y = ratio of leading coefficients?
Numerator degree = denominator degree.
What is the condition for no horizontal asymptote?
Numerator degree > denominator degree.
What is the formula for combinations (binomial coefficients)?
$\binom{n}{k} = \frac{n!}{k!(n-k)!}$