Difference between revisions of "2022 AMC 10B Problems/Problem 21"

Revision as of 14:50, 25 November 2022

Problem

Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$ , the remainder is $x+2$ , and when $P(x)$ is divided by the polynomial $x^2+1$ , the remainder is $2x+1$ . There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?

Solution 1 (Experimentation)

Given that all the answer choices and coefficients are integers, we hope that $P(x)$ has positive integer coefficients.

Throughout this solution, we will express all polynomials in base $x$ . E.g. $x^2 + x + 1 = 111_{x}$ .

We are given:

$111a + 12 = 101b + 21 = P(x)$ .

We add $111$ and $101$ to each side and balance respectively:

$111(a - 1) + 123 = 101(b - 1) + 122 = P(x)$

We make the units digits equal:

$111(a - 1) + 123 = 101(b - 2) + 223 = P(x)$

We now notice that:

$111(a - 11) + 1233 = 101(b - 12) + 1233 = P(x)$ .

Therefore $a = 11_{x} = x + 1$ , $b = 12_{x} = x + 2$ , and $P(x) = 1233_{x} = x^3 + 2x^2 + 3x + 3$ . $3$ is the minimal degree of $P(x)$ since there is no way to influence the $x$ ‘s digit in $101b + 21$ when $b$ is an integer. The desired sum is $1^2 + 2^2 + 3^2 + 3^2 = \boxed{\textbf{(E)} \ 23}$

P.S. The 4 computational steps can be deduced through quick experimentation.

~ numerophile

Solution 2

Let $P(x) = Q(x)(x^2+x+1) + x + 2$ , then $P(x) = Q(x)(x^2+1) + xQ(x) + x + 2$ , therefore $xQ(x) + x + 2 \equiv 2x + 1 \pmod{x^2+1}$ , or $xQ(x) \equiv x-1 \pmod{x^2+1}$ . Clearly the minimum is when $Q(x) = x+1$ , and expanding gives $P(x) = x^3+2x^2+3x+3$ . Summing the squares of coefficients gives $\boxed{\textbf{(E)} \ 23}$

~mathfan2020

Solution 3

Let $P(x) = (x^2+x+1)Q_1(x) + x + 2$ , then $P(x) = (x^2+1)Q_1(x) + xQ_1(x) + x + 2$

Also $P(x) = (x^2+1)Q_2(x) + 2x + 1$

We infer that $Q_1(x)$ and $Q_2(x)$ have same degree, we can assume $Q_1(x) = x + a$ , and $Q_2(x) = x + b$ , since $P(x)$ has least degree. If this cannot work, we will try quadratic, etc.

Then we get: $(x^2+1)(Q_1(x) - Q_2(x)) + xQ_1(x) - x + 1 = 0$

The constant term gives us: $(Q_1(x) - Q_2(x)) + 1 = 0$

So $Q_1(x) - Q_2(x) = -1$

Substituting this in gives: $-(x^2+1) + xQ_1(x) - x + 1 = 0$

Solving this equation, we get $Q_1(x) = x + 1$

Plugging this into our original equation we get $P(x) = x^3 + 2x^2 + 3x + 3$

Verify this works with $P(x) = (x^2+1)Q_2(x) + 2x + 1$

Therefore the answer is $1^2 + 2^2 + 3^2 + 3^2 = \boxed{\textbf{(E)} \ 23}$

~qgcui

Solution 4 (undetermined coefficients)

Notice that we cannot have the quotients equal to some constants, since the same constant will yield different constant terms for $P(x)$ (which is bad) and different constants will yield different first coefficients (also bad). Thus, we try setting the quotients equal to linear terms (for minimizing degree).

Let $P(x)=(x^2+x+1)(ax+b)+(x+2)$ and $P(x)=(x^2+1)(ax+c)+(2x+1)$ . The quotients have the same $x$ coefficient, since $P(x)$ must have the same $x^3$ coefficient in both cases. Expanding, we get $P(x)=ax^3+(a+b)x^2+(a+b+1)x+(b+2)$ and $P(x)=ax^3+cx^2+(a+2)x+(c+1).$

Equating coefficients, we get $b+2=c+1$ , $a+b+1=a+2$ , and $a+b=c$ . From the second equation, we get $b=1$ , then substituting into the first, $c=2$ . Finally, from $a+b=c$ , we have $a=1$ . Now, $P(x)=(x^2+x+1)(ax+b)+(x+2)=(x^2+x+1)(x+1)+(x+2)=x^3+2x^2+3x+3$ and our answer is $1^2+2^2+3^2+3^2=\boxed{\textbf{(E)} \ 23}.$

~MathHayden

Solution 5: Quick (But not quicker than 2)

We construct the following equations in terms of $P(x)$ and the information given by the problem: $\textbf{(1) } P(x)=(x^2+x+1)\cdot Q(x)+x+2$ $\textbf{(2) } P(x)=(x^2+1)\cdot R(x)+2x+1$ Upon inspection, $Q(x)$ and $R(x)$ cannot be constant, so the smallest possible degree of $P(x)$ is $3,$ and both $Q(x)$ and $R(x)$ are linear.

Let $Q(x)=x-q$ and $R(x)=x-r.$ We know there will be values for $q$ and $r$ that make the below equation hold, so we can assume that $P(x)$ has a leading coefficient of $1$ .

Substituting these values in, and setting $\textbf{(1)}$ and $\textbf{(2)}$ equal to each other, $(x^2+x+1)(x-q)+x+2=(x^2+1)(x-r)+2x+1.$ We plug in $x=0$ , yielding $r+1=q.$ Substituting this value into the above equation, $(x^2+x+1)(x-r-1)+x+2=(x^2+1)(x-r)+2x+1.$ Letting $x=1,$ we conclude that $r=-2,$ so $R(x)=x+2.$ Therefore, $P(x)=(x^2+1)(x+2)+2x+1 = x^3+2x^3+3x+3.$ The requested sum is $1^2+2^2+3^2+3^2=\boxed{\textbf{(E) }23}$

-Benedict T (countmath1)

Video Solutions

https://youtu.be/yGUur4vP_6k

~ ThePuzzlr

https://youtu.be/ELdhkqVyB9E

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution by OmegaLearn using Circular Tangency

https://youtu.be/HdrbPiZHim0

~ pi_is_3.14

@@ Line 89: / Line 89: @@
 Upon inspection, <math>Q(x)</math> and <math>R(x)</math> cannot be constant, so the smallest possible degree of <math>P(x)</math> is <math>3,</math> and both <math>Q(x)</math> and <math>R(x)</math> are linear.
-Let <math>Q(x)=x-q</math> and <math>R(x)=x-r.</math> We know there will be values for <math>q</math> and <math>r</math> that make the below equation hold, so our assumption that <math>P(x)</math> has a leading coefficient of <math>1</math> is true.
+Let <math>Q(x)=x-q</math> and <math>R(x)=x-r.</math> We know there will be values for <math>q</math> and <math>r</math> that make the below equation hold, so we can assume that <math>P(x)</math> has a leading coefficient of <math>1</math>.
 Substituting these values in, and setting <math>\textbf{(1)}</math> and <math>\textbf{(2)}</math> equal to each  other,

2022 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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