The Remainder Theorem states that when p(x) is divided by (x − c), the remainder equals: — p(c).The Remainder Theorem: dividing p(x) by (x − c) gives remainder r = p(c). This lets you find remainders without full division.
Is (x − 3) a factor of p(x) = x³ − 27? — Yes, because p(3) = 0.p(3) = 27 − 27 = 0. By the Factor Theorem, since p(3) = 0, (x − 3) is indeed a factor.
What is the remainder when x³ + 2x − 1 is divided by (x − 1)? — 2.p(1) = 1 + 2 − 1 = 2. By the Remainder Theorem, the remainder is 2.
What degree is the quotient when a degree-4 polynomial is divided by a degree-1 polynomial? — 3.When dividing a degree-4 polynomial by a degree-1 polynomial (like x − c), the quotient has degree 4 − 1 = 3.
In synthetic division of p(x) ÷ (x − 3), what number goes in the box? — 3.In synthetic division by (x − c), the value c goes in the box. For (x − 3), c = 3 goes in the box.
What does it mean if p(c) = 0? — (x − c) is a factor of p(x).By the Factor Theorem (corollary to Remainder Theorem): p(c) = 0 if and only if (x − c) is a factor of p(x).
The Factor Theorem is a special case of the Remainder Theorem. Which condition makes a factor? — p(c) = 0.The Factor Theorem states (x − c) is a factor of p(x) if and only if p(c) = 0 (remainder = 0).
If p(x) = x³ + 6x² + 11x + 6, and p(−1) = 0, what is a factor? — (x + 1).Since p(−1) = 0, by the Factor Theorem (x − (−1)) = (x + 1) is a factor.
Synthetic division is especially efficient when dividing by polynomials of the form: — (x − c).Synthetic division is a shortcut specifically designed for division by linear factors of the form (x − c), where the leading coefficient is 1.
What is the first step when factoring any polynomial? — Factor out the GCF.Always pull out the Greatest Common Factor (GCF) first. This simplifies the remaining polynomial and makes other techniques easier to apply.
Which polynomial has x = 1 as a double root? — x² − 2x + 1.x² − 2x + 1 = (x − 1)². The exponent 2 means x = 1 is a double root.
Which is a factor of x⁴ − 1? — Both (x² + 1) and (x² − 1).x⁴ − 1 = (x² − 1)(x² + 1) by difference of squares. Both are factors.
What technique would you use first to factor 2x⁴ − 8x? — Factor out 2x.GCF = 2x. Factor out 2x first: 2x(x³ − 4). Then check if x³ − 4 factors further.
What is the factored form of x³ + 6x² + 9x? — x(x + 3)².GCF = x: x(x² + 6x + 9) = x(x + 3)².
A vertical asymptote occurs at x = a when: — (x − a) remains in the denominator after simplification.A vertical asymptote occurs when (x − a) is a factor of the denominator that does NOT cancel with the numerator. If it cancels, there is a hole instead.
A "hole" in a rational function occurs when: — A factor cancels from both numerator and denominator.A hole (removable discontinuity) occurs when a factor cancels from both numerator and denominator. The function is undefined at that x-value but the limit exists.
What is the horizontal asymptote of f(x) = (3x² + 2)/(x² − 5)? — y = 3.Degrees are equal (both 2). Horizontal asymptote = ratio of leading coefficients = 3/1 = 3, so y = 3.
What is the horizontal asymptote of f(x) = (2x + 1)/(x² − 1)? — y = 0.The degree of the numerator (1) is less than the degree of the denominator (2). When deg(p) < deg(q), the horizontal asymptote is y = 0.
For f(x) = 5/(x² − 4), what are the vertical asymptotes? — x = ±2.Set denominator = 0: x² − 4 = 0 → x = ±2. Neither cancels with the numerator (5), so both are vertical asymptotes.
What is the x-intercept (if any) of f(x) = (x − 4)/(x + 2)? — x = 4.x-intercepts occur where the numerator = 0 (and denominator ≠ 0). x − 4 = 0 → x = 4. Check: denominator = 4+2 = 6 ≠ 0. ✓
What is the horizontal asymptote of f(x) = (4x³ − 1)/(2x³ + x)? — y = 2.Equal degrees (both 3). Horizontal asymptote = leading coefficients ratio = 4/2 = 2. So y = 2.
Which feature of the graph of a rational function occurs at x = c if p(c) ≠ 0 and q(c) = 0? — Vertical asymptote.When q(c) = 0 and p(c) ≠ 0, the factor (x−c) is in the denominator but not in the numerator (no cancellation), giving a vertical asymptote at x = c.
To multiply two rational expressions, you: — Multiply numerators and multiply denominators.(a/b) · (c/d) = (ac)/(bd). Multiply numerators together and denominators together, then simplify by canceling common factors.
To divide two rational expressions (a/b) ÷ (c/d), you: — Multiply a/b by d/c.Division by a fraction = multiplication by its reciprocal: (a/b) ÷ (c/d) = (a/b) · (d/c) = (ad)/(bc).
What is the LCD of (1/x²) + (1/x)? — x².LCD of x² and x is x² (the highest power). Rewrite: 1/x² + x/x² = (1+x)/x².
Add: 3/(x+2) + x/(x−1). What is the resulting denominator? — (x+2)(x−1).The LCD is the product of the distinct denominators: (x+2)(x−1). Note (x+2)(x−1) = x²+x−2, so choice 3 is also equal — but the factored form is (x+2)(x−1).
When must you state domain restrictions for rational expressions? — Whenever the denominator can equal zero.Domain restrictions must be stated whenever the denominator could equal zero, because the expression is undefined at those values — even if they cancel during simplification.
Which expression is in fully simplified (lowest terms) form? — (x−1)/(x+2).(x−1)/(x+2) has no common factors (x−1 and x+2 share no common polynomial factor). The others can be simplified further.