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Difference between revisions of "2008 iTest Problems/Problem 85"

(Solution to Problem 85 (credit to tenniskidperson3 and official solution) -- system with a hidden form)
 
m (→‎Solution)
 
Line 10: Line 10:
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
p(x) &= (x^2 + ax + c)(x^2 + bx + d) \\
 
p(x) &= (x^2 + ax + c)(x^2 + bx + d) \\
−
&= x^4 + (a+d)x^3 + (ab+c+d)x^2 + (ad+bc)x + cd \\
+
&= x^4 + (a+b)x^3 + (ab+c+d)x^2 + (ad+bc)x + cd \\
−
&= x^4 + 15x^3 + 78x + 160x + 96
+
&= x^4 + 15x^3 + 78x^2 + 160x + 96
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 
Factoring <math>p(x)</math> with the [[Rational Root Theorem]] results in <math>(x+4)(x+4)(x+1)(x+6)</math>. By the [[Fundamental Theorem of Algebra]], we know that <math>x+4, x+4, x+1, x+6</math> are all the linear factors of the polynomial, so the quadratic factors can only be multiplied from these linear factors.
 
Factoring <math>p(x)</math> with the [[Rational Root Theorem]] results in <math>(x+4)(x+4)(x+1)(x+6)</math>. By the [[Fundamental Theorem of Algebra]], we know that <math>x+4, x+4, x+1, x+6</math> are all the linear factors of the polynomial, so the quadratic factors can only be multiplied from these linear factors.

Latest revision as of 22:18, 15 August 2020

Problem

Let $(a,b,c,d)$ be a solution to the system \begin{align*}a+b&=15,\\ab+c+d&=78,\\ad+bc&=160,\\cd&=96.\end{align*} Find the greatest possible value of $a^2+b^2+c^2+d^2$.

Solution

Note that when multiplying quadratics, terms add up similar to the equations of a system, so let \begin{align*} p(x) &= (x^2 + ax + c)(x^2 + bx + d) \\ &= x^4 + (a+b)x^3 + (ab+c+d)x^2 + (ad+bc)x + cd \\ &= x^4 + 15x^3 + 78x^2 + 160x + 96 \end{align*} Factoring $p(x)$ with the Rational Root Theorem results in $(x+4)(x+4)(x+1)(x+6)$. By the Fundamental Theorem of Algebra, we know that $x+4, x+4, x+1, x+6$ are all the linear factors of the polynomial, so the quadratic factors can only be multiplied from these linear factors.


There are only two possible distinct groupings (not counting rearrangements) -- $(x^2 + 8x + 16)(x^2 + 7x + 6)$ and $(x^2 + 5x + 4)(x^2 + 10x + 24)$. In the first case, $a^2 + b^2 + c^2 + d^2 = 405$, and in the second case, $a^2 + b^2 + c^2 + d^2 = 717$. The largest of the two options is $\boxed{717}$.

See Also

2008 iTest (Problems)
Preceded by:
Problem 84 		Followed by:
Problem 86
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