polynomials

synthetic division reminders

• binomial must be x^1 ± ?

• ALL coefficients must be listed

definition of factor

when dividing, if the remainder is 0, the divisor is a factor of the dividend

remainder theorem

If P(x) is a polynomial and “a” is a number, and if P(x) is dividend by x-a, then the remainder is P(a)

factor theorem

If P(x) is a polynomial, then x-c is a factor of P(x) if and only if P(c) = 0

determine whether the given binominal is a factor of P(x) and justify: plug in method

Based on the Factor Theorem, since P(#) = 0, x-# is a factor.

determine whether the given binominal is a factor of P(x) and justify: synthetic division method

According to the remainder theorem, P(#)=0. Based on the factor theorem, x-# is a factor since P(#)=0.

if the zeros are fractions, what do the numerators/denominators represent

• numerator: factors of “c”

• denominator: factors of “a”

rational zeros/root theorem

if a polynomial function has integral coefficients (may need to sweep), and it has a rational zero p/q, where p and q are relatively prime, then p is a factor of the constant term and q is a factor of the leading coefficient

finding zeros using RRT

1. find p’s and q’s

2. divide p/q’s to find PRRs

3. first plug in 1, and plug in other solutions using synthetic division

4. factor depressed polynomial by grouping

5. set factors =0 and solve

corollary of the rationals zeros theorem

"Leading coefficient = 1? Then rational zeros are whole numbers."

Use it when: the x² (or highest) term has no number in front of it → possible rational zeros are just the factors of the constant

T/F: In P(x) = x³ + x² + 11x - 6, a possible zero is ½

FALSE → zeros should be integers (corollary of the rationals zeros theorem)

fundamental theorem of algebra

"Every polynomial has at least one zero, no matter what."

Use it when: you need to prove/know that a solution always exists — even if you have to go into complex/imaginary numbers to find it.

corollary of fundamental theorem of algebra

"Degree of the polynomial = exact number of zeros."

Use it when: you need to know how many total solutions a polynomial has. x³ → 3 zeros, x⁵ → 5 zeros, etc. (some may repeat or be imaginary)

complex conjugate theorem

"Imaginary zeros come in pairs: a+bi always brings a-bi with it." Use it when: you're finding zeros and you get an imaginary number — you automatically get its "mirror" for free. Also works backwards: if you're building a polynomial from its zeros, always include both.

irrational conjugate theorem

"Radical zeros come in pairs: a+√b always brings a−√b with it." Use it when: you're finding zeros and you get square roots. Get one irrational root → you get its conjugate free.

descartes rule of signs

• for pos roots: count # of sign changes, subtract by 2 until =0

• for neg roots: make x neg for terms w/ odd degrees, count # sign changes, subtract by 2 until =0

• for img: add pos and neg roots, subtract by highest degree

• for zero: degree of factored x

Quadratic — roots

0, 1, or 2

Cubic — roots

1 or 3

Quartic — roots

0, 2, or 4

Quintic — roots

1, 3, or 5

Quadratic — shape (a>0)

U‑shape

Cubic — shape (a>0)

S‑shape

Quartic — shape (a>0)

W‑shape

Quintic — shape (a>0)

increasing multi‑turn curve

Quadratic — shape (a

upside‑down U

Cubic — shape (a

inverted S

Quartic — shape (a

upside‑down W

Quintic — shape (a

decreasing multi‑turn curve

All polynomials — domain

(-∞, ∞)

Quadratic range (a>0)

y ≥ k

Quadratic range (a

y ≤ k

Cubic range

(-∞, ∞)

Quartic range (a>0)

y ≥ k

Quartic range (a

y ≤ k

Quintic range

(-∞, ∞)