Coe Mod
help with number theory proof?
prove: Suppose f(x) = a(subn)x^n + a(subn−1) x^(n−1) + . . . + a(sub0) is a polynomial of degree n > 0
with integer coefficients. Let a, b and m be integers with m > 0. If a ≡ b (mod m), then f(a) ≡ f(b) (mod m).
Intuitively, it’s true because polynomials are composed entirely of multiplications, additions, and subtractions. These operations preserve equality in moduli. So, if a ≡ b (mod m), then:
a + c ≡ b + c
a – c ≡ b – c
ac ≡ bc
If we combine multiple additions, subtractions, and multiplications, we obtain the set of all polynomials, so for any polynomial:
a ≡ b ===> f(a) ≡ f(b)
More formally, we can prove this by induction on the degree of f. Suppose f is of degree 0. Then, for some constant c:
f(x) = c … for all x
Therefore, if a ≡ b, then f(a) = c = f(b) ===> f(a) ≡ f(b). Therefore, we have our base case. Suppose it works for all polynomials of degree k. Let f be a polynomial of degree k + 1, i.e.
f(x) = a_(k + 1)x^(k + 1) + a_k x^k + … + a_1 x + a_0
= x(a_(k + 1)x^k + a_k x^(k – 1) + … + a_1) + a_0
for some a_0, …, a_(k + 1), where a_(k + 1) is non-zero. Let g(x) = a_(k + 1)x^k + a_k x^(k – 1) + … + a_1. Then g(x) is of degree k, since a_(k + 1) does not equal 0, and:
f(x) = x g(x) + a_0
By the induction hypothesis, we know that:
b ≡ c ===> g(b) ≡ g(c)
===> b g(b) ≡ b g(c)
===> b g(b) ≡ c g(c)
===> b g(b) + a_0 ≡ c g(c) + a_0
===> f(b) ≡ f(c)
Therefore, by induction, b ≡ c ===> f(b) ≡ f(c) for all polynomials f.
CALL OF DUTY4 Mod COE
| | Amy Coe Mod Baby Pink Plush Butterfly 2 Sided NWOT $24.99 |
| | Amy Coe BLUE ROARY LION SECURITY BLANKET Mod Lovey $22.99 |
| | Amy Coe Baby Mod Peanut Green Elephant Plush Lovey Toy $19.99 |
| | NWT Boutique AMY COE Limited Edition MOD BABY Plush Toy Elephant $19.99 |
| | Girls AMY COE Pink Plush Mod BUTTERFLY Lovey $16.99 |
| | AMY COE LOVE FLOWER MOD BABY PINK SECURITY BLANKET LOVEY $14.99 |
| | Amy Coe Mod Green Elephant Plush Toy Washable $14.99 |
| | AMY COE MOD BABY Green Elephant Plush Stuffed Animal Lovey Toy Soft 7″ Long NWT $14.96 |
| | Amy Coe Mod Baby Pink Butterfly Plush Lovey Stuffed Animal Soft Toy $13.99 |
| | AMY COE Mod Peanut GREEN ELEPHANT Plush Lovey Baby Toy $13.99 |
| | Amy Coe Green Elephant Mod Safari Plush Toy $12.99 |
| | Amy Coe Mod Green Plush Elephant Peanut Baby Toy 8″ EUC $12.99 |
| | Amy Coe Elephant Mod Peanut Green Baby Plush Lovey Stuffed Animal Toy Crib 7″ $12.50 |
| | Green AMY COE Mod Safari Peanut ELEPHANT Baby Lovey Toy $11.99 |
| | CUTE AMY COE SLEEP/AWAKE PINK PLUSH MOD BABY BUTTERFLY $11.95 |
| | Amy Coe Green Mod Elephant Peanut stuffed animal plush baby infant toy Target $11.95 |
| | AMY COE Mod Plush Green Elephant Baby Toy Lovely Stuffed Animal 8″ Super Soft $11.95 |
| | Amy Coe Target MOD PEANUT Green Elephant Plush Stuffed Animal Lovey $10.91 |
| | Amy Coe MOD Peanut Green Elephant Plush Baby Toy Lovey 8″ $9.95 |
| | Amy Coe Pink Mod Butterfly Awake Asleep Plush Lovey Toy $8.99 |
| | 10″ Amy Coe Target Green Elephant MOD PEANUT Safari Jungle Plush Baby Toy $8.99 |
| | AMY COE BABY Mod Peanut GREEN Plush ELEPHANT Lovey TOY Stuffed ROOM DECOR Target $8.99 |
| | Amy Coe™ Mod Baby Flutter Pillow – Pink The colors and shape make it modern enough to please mom; the adorable details are 100% pure baby. Place this pretty accent pillow from the Amy Coe limited edition Mod Baby collection on your nursery rocker or in the reading chair. The cotton shell features a white appliqué butterfly; mini white dots on a chocolate background enliven the border and reverse side. Best of all, this clever design is… |
| | Amy Coe™ Flutter Luxe Microfleece Sweet Spots Blanket – Pink So irresistibly soft you’ll wish it was big enough for your own bed, this scrumptious blanket is made just for your little sweetie to snuggle. One plush side features a generous graphic print of pale pink spots on a rosy pink background; the other displays the same hues in reverse. Complements the Sprinkle Spots Crib Sheet and Flutter Bedding Set from the Amy Coe limited edition Mod Baby collectio… |
| | Coll Algebra Graphs& Mod& Mymthlb& Coe Stk Pkg $126.67 … |
| | Alg& Trig: Graphs& Mod W/MML& Ssm& Cpn& Coe Pkg $153.27 … |
| | Amy Coe Peanut Plush Elephant – Green … |
| | Limited Edition Amy Coe Mod Safari Roary Lion Lovey Blankie Blue Baby Security Blanket A rare find! A Limited Edition Blankette by Amy Coe from the Mod Safari Collection! Super soft and cuddly for baby in a light baby blue color with cute Roary lion head…. |
