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Chapter Review

Quadratic Equations and Polynomials

Quadratic Roots · Polynomials and Synthetic Division · Partial Fractions

Quadratic Formula and the Discriminant

The quadratic formula solves any equation using the discriminant to determine whether roots are real, repeated, or complex.

Key Points

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    : two distinct real roots; if is a perfect square (rational ), roots are rational
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    : one repeated real root at (the vertex)
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    : two complex conjugate roots
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    Irrational roots come in conjugate pairs when are rational
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    Always compute first before attempting to find root values
Formula

Vieta's Formulas

Vieta's formulas relate the sum and product of quadratic roots directly to the coefficients, bypassing the need to solve for individual roots.

Key Points

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    Sum of roots:
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    Product of roots:
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    For a monic quadratic (): sum , product
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    Construct equation from roots:
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Formula

Cube Roots of Unity

The three solutions to are where and are complex conjugates equally spaced at 120° on the unit circle.

Key Points

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    , found from
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    Sum property:
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    Cyclic property: , so reduce exponents mod 3
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    and are complex conjugates of each other
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    Factorization:
Formula

Fourth Roots of Unity

The four solutions to are , obtained by factoring .

Key Points

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    Two real roots () and two imaginary roots ()
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    Sum of all four roots is zero
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    Product of all four roots is (unlike cube roots where product is )
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    Spaced 90° apart on the unit circle
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    To find : multiply each fourth root of unity by

Polynomial Division Algorithm

Any polynomial divided by yields a quotient and remainder such that , with deg() < deg().

Key Points

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    Both polynomials must be in standard form (descending powers) before dividing
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    Dividing degree- by degree- gives quotient of degree
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    If , the divisor is a factor of the dividend
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    Insert placeholder zeros for any missing powers of
Formula

Synthetic Division

Synthetic division is a streamlined coefficient-only method for dividing a polynomial by a linear factor , using multiply-and-add steps.

Key Points

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    Only works for linear divisors with leading coefficient 1
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    For divisor , use (flip the sign)
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    Drop the leading coefficient, then repeatedly multiply by and add
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    The last number in the bottom row is the remainder; the rest are quotient coefficients
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    Insert 0 for any missing powers in the dividend

Remainder and Factor Theorems

The Remainder Theorem states equals the remainder when dividing by ; the Factor Theorem follows as the special case where means is a factor.

Key Points

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    Remainder Theorem: — evaluate instead of dividing
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    Factor Theorem: is a factor
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    Use to find unknown coefficients: set and solve
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    Depressed equation: after removing factor , solve the lower-degree quotient for remaining roots
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    Repeated synthetic division reduces degree by 1 each time
Formula

Partial Fractions — Distinct Linear Factors

A proper rational fraction with distinct linear factors in the denominator decomposes into a sum of simple fractions, each with a constant numerator over one linear factor.

Key Points

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    Check proper: deg(numerator) < deg(denominator); if not, long-divide first
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    Each distinct factor contributes
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    Clear denominators, then substitute roots to isolate each constant
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    Cover-up shortcut: cover a factor in the denominator and evaluate the rest at that root
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    Always verify by recombining the partial fractions
Formula

Partial Fractions — Repeated Linear Factors

A repeated factor in the denominator generates partial fraction terms with ascending powers from 1 to , each with a constant numerator.

Key Points

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    Cannot skip powers: needs terms for , , and
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    Substitute to find the highest-power constant first
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    Use equating coefficients for the remaining constants
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    Watch for disguised repeats: e.g.
Formula

Partial Fractions — Irreducible Quadratic Factors

An irreducible quadratic factor () in the denominator requires a linear numerator , not just a constant.

Key Points

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    Numerator degree must be one less than denominator factor degree
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    Irreducibility check: discriminant
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    Repeated irreducible quadratic produces terms, each with a linear numerator
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    Use substitution for linear roots first, then equate coefficients for quadratic unknowns
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    Hybrid approach: substitution + equating coefficients is most efficient
Formula

Formulas

Quadratic Formula

Solves any quadratic equation using coefficients a, b, c.

Discriminant

Determines nature of roots: positive → 2 real, zero → 1 repeated, negative → complex.

Vieta's — Sum and Product

Relates roots to coefficients without solving the equation.

Cube Roots of Unity

The three cube roots of 1 sum to zero; ω³ = 1.

Remainder Theorem

Remainder of P(x) ÷ (x−c) equals P(c).

Factor Theorem

A binomial (x−c) is a factor if and only if P(c) = 0.

Partial Fractions — Distinct Linear

Each distinct linear factor gets a constant numerator.

Partial Fractions — Irreducible Quadratic

Quadratic denominator factor requires a linear numerator Ax+B.