5. Let the polynomial $p$ be defined by $p(x)=x^{4}+3 x^{3}+a x^{2}+b x+7$. Determine the numbers $a$ and $b$ such that the line $z$ with the equation $y=2 x+7$ intersects the graph of the polynomial $p$ at points with abscissas 1 and -4.

(8 points)

## 12th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 21, 2012

## Solutions to Problems and Scoring Guide

## A contestant who arrives at the solution by any correct method (even if not anticipated by the scoring guide) receives all possible points.

A correct method is considered any procedure that:

- sensibly takes into account the problem statement,
- leads to the solution of the problem,
- is mathematically correct and complete.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the contestant generally receives all the points anticipated. However, if the solution is guessed (and cannot be arrived at without calculation), such a solution is scored with 0 points.

If a contestant has only partially solved the problem, and the correct procedures are visible but the path to the final solution is not clear, the contestant cannot receive more than half of the possible points.

The asterisk '*' next to the points means that the contestant can receive the point or points for a correct procedure, even if the calculation is incorrect.

## First Year
5. I. method:

We write the equation $2 x+7=x^{4}+3 x^{3}+a x^{2}+b x+7$. We rearrange the equation and get $x^{4}+3 x^{3}+a x^{2}+(b-2) x=0$. We substitute $x=1$ into the equation and get $1+3+a+b-2=0$ or $a+b+2=0$. We substitute $x=-4$ into the equation and get $256-192+16 a-4 b+8=0$ or $4 a-b+18=0$. We solve the system of equations and get $a=-4$ and $b=2$.

Writing the equality $2 x+7=x^{4}+3 x^{3}+a x^{2}+b x+7 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ Rearranged equation $x^{4}+3 x^{3}+a x^{2}+(b-2) x=0 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$
Writing the equation $a+b+2=0$ ..... $1^{*}$ point
Writing the equation $4 a-b+18=0$ ..... $1^{*}$ point
Solving the system of equations ..... $2^{*}$ points
Solution $a=-4$ ..... 1 point
Solution $b=2$ ..... 1 point
II. method:
We calculate the coordinates of the intersection points of the line and the graph of the polynomial $(1,9)$ and $(-4,-1)$. We substitute the coordinates of both points into the formula for the polynomial. We get the equations: $a+b=-2,8 a-2 b=-36$.
We solve the system of equations and get $a=-4$ and $b=2$.
Calculating the ordinates of the points $(1,9)$ and $(-4,-1)$ ..... $1+1$ points
Writing the equations $a+b=-2,8 a-2 b=-36 \ldots$ ..... $1^{*}+1^{*}$ points
Solving the system of equations ..... $2^{*}$ points
Solution $a=-4$ ..... 1 point
Solution $b=2$ ..... 1 point