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1.11 · Hole vs VA — Core Rule
h(x) = f(x)/g(x). If g(c) = 0, what are the TWO possible features at x = c?
What does it depend on?
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1.11 · Hole — When Does It Occur?
When does a rational function have a HOLE at x = c?
Think: numerator and denominator share a factor
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1.11 · VA — When Does It Occur?
When does a rational function have a VERTICAL ASYMPTOTE at x = c?
Think: denominator only
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1.11 · Hole vs VA — How to Tell
h(x) = (x−2)(x+3) / (x+3)(x−5). Is x = −3 a hole or a VA?
(x+3) appears in both numerator and denominator
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1.11 · Zeros of Rational Function
h(x) = (x+3) / x(x+5). What are the zeros of h(x)?
Zeros come from the numerator only
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1.11 · Domain of Rational Function
h(x) has a hole at x = 4 and a VA at x = −2. What is the domain?
Exclude ALL discontinuities
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1.11 · HA Rule — deg(num) < deg(denom)
What is the horizontal asymptote when deg(num) < deg(denom)?
y = ?
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1.11 · HA Rule — Equal Degrees
What is the HA when deg(num) = deg(denom)?
Leading coefficients: numerator a, denominator b
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✓ Hole — When Does It Occur?
A hole occurs at x = c when (x − c) is a factor of BOTH numerator AND denominator. The factor cancels → removable discontinuity (open circle on graph).
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✓ Hole vs VA — Core Rule
Either a HOLE (if the factor also cancels from the numerator) or a VERTICAL ASYMPTOTE (if it doesn't cancel). Must check the numerator to determine which.
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✓ Hole vs VA — How to Tell
HOLE at x = −3. The factor (x + 3) appears in both numerator and denominator — it cancels. The simplified function is (x − 2)/(x − 5). x = 5 would be the VA.
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✓ VA — When Does It Occur?
A vertical asymptote occurs at x = c when (x − c) is a factor of the denominator ONLY — it does NOT cancel with the numerator. Function → ±∞ as x → c.
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✓ Domain of Rational Function
(−∞, −2) ∪ (−2, 4) ∪ (4, ∞). Exclude ALL x-values where the original denominator = 0, including holes.
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✓ Zeros of Rational Function
Zero at x = −3 only. x = 0 and x = −5 are VAs (denominator zeros), not function zeros.
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✓ HA Rule — Equal Degrees
y = a/b (ratio of leading coefficients). The function levels off at this value as x → ±∞.
Example: (3x²+1)/(2x²−5) → y = 3/2
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✓ HA Rule — deg(num) < deg(denom)
y = 0. When the denominator grows faster than the numerator, the function shrinks toward zero.
Example: (x+1)/(x²+3) → y = 0
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1.11 · Slant Asymptote — When?
When does a rational function have a SLANT asymptote?
Think about the degree difference
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1.11 · Slant Asymptote — How to Find
How do you find the equation of a slant asymptote?
deg(num) = deg(denom) + 1
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1.11 · Long Division — Answer Form
When dividing f(x) by g(x), what form does the answer take?
f(x)/g(x) = q(x)·g(x) + r(x)
q = quotient, r = remainder
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1.11 · Pascal's Triangle — Row to Use
To expand (a + b)ⁿ, which row of Pascal's Triangle do you use?
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1.11 · Binomial Theorem — Powers Pattern
In (a + b)⁵, what happens to the powers of a and b term by term?
Start at degree 5 for a, degree 0 for b
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1.11 · Binomial Theorem — Negative b
Expand (2x − 1)⁴. What is the sign pattern of the terms?
b = −1. Signs alternate: + − + − +
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1.11 · C(n,r) — Finding One Term
What is the coefficient of x³ in the expansion of (x − 3)⁸?
C(8,5) since (−3) appears 5 times
C(8,5) = 56
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1.11 · Reading a Graph — Hole
A graph shows an OPEN CIRCLE at x = 4. What does this mean for the factored equation?
What goes in numerator? Denominator?
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✓ Slant Asymptote — How to Find
Use LONG DIVISION. Divide numerator by denominator. The QUOTIENT (ignoring the remainder) is the slant asymptote y = mx + b.
The remainder → 0 as x → ±∞
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✓ Slant Asymptote — When?
When deg(numerator) = deg(denominator) + 1. The numerator degree is exactly ONE MORE than the denominator. No horizontal asymptote in this case.
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✓ Pascal's Triangle — Row to Use
Use Row n for (a+b)ⁿ. For (a+b)⁵ use Row 5: coefficients 1, 5, 10, 10, 5, 1. Row 0 is just '1' at the top.
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✓ Long Division — Answer Form
f(x)/g(x) = q(x) + r(x)/g(x), where q is the quotient and r is the remainder. Degree of r must be less than degree of g.
Example: (6x²+x+5)/(2x+1) = (3x−1) + 6/(2x+1)
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✓ Binomial Theorem — Negative b
Signs alternate + − + − + because b = −1 gives (−1)⁰=1, (−1)¹=−1, etc. Result: 16x⁴ − 32x³ + 24x² − 8x + 1
Always apply the negative with b!
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✓ Binomial Theorem — Powers Pattern
Powers of a: 5, 4, 3, 2, 1, 0. Powers of b: 0, 1, 2, 3, 4, 5. They always add up to n = 5.
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✓ Reading a Graph — Hole
(x − 4) goes in BOTH numerator AND denominator. The common factor cancels, producing the open circle. Distinct from a zero (numerator only) and VA (denominator only).
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✓ C(n,r) — Finding One Term
C(8,5) · x³ · (−3)⁵ = 56 · x³ · (−243) = −13,608x³. Coefficient = −13,608.