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- Feb 14, 2012
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The polynomial $g(x)$ is cubic. What is the largest value of $a$ if $f_1(x)=x^2+(a-29)x-a$ and $f_2(x)=2x^2+(2a-43)x+a$ are both factors of $g(x)$?
The polynomial $g(x)$ is cubic. What is the largest value of $a$ if $f_1(x)=x^2+(a-29)x-a$ and $f_2(x)=2x^2+(2a-43)x+a$ are both factors of $g(x)$?
Thanks for participating, Opalg!If the quadratic polynomials $f_1(x)$ and $f_2(x)$ both divide the cubic polynomial $g(x)$ then they must have a factor in common. Any common factor of $f_1(x)$ and $f_2(x)$ must also be a factor of $2f_1(x) - f_2(x) = -15\bigl(x+\frac15a\bigr)$. Therefore $f_1(x) = x^2+(a-29)x-a = \bigl(x+\frac15a\bigr)(x-5)$ (the second factor has to be $x-5$ in order to make the constant term equal to $a$). Compare the coefficients of $x$ to see that $a-29 = \frac15a - 5$, from which $a=30$.
A similar calculation using $f_2(x)$ instead of $f_1(x)$ confirms the solution $a=30$.
I hope I am not missing something here. The question asks for the largest value of $a$, but as far as I can see there is only the one possible value for $a$.
$f_1(k)=k^2+(a-29)k-a=0$ | $f_2(k)=2k^2+(2a-43)k+a=0$ |
$k^2+(a-29)k-a=0$ | $2k^2+(2a-43)k+a=0$---(2) |
$2k^2+2(a-29)k-2a=0$---(1) |
$f_1(k)=k^2+(-5k-29)k+5k=0$ | $f_2(k)=2k^2+(2(-5k)-43)k-5k=0$ |
$k^2-5k^2-29k+5k=0$ $-4k^2-24k=0$ $k(k+6)=0$ $k=0$ or $k=-6$ | $2k^2-10k^2-43k-5k=0$ $-8k^2-48k=0$ $k(k+6)=0$ $k=0$ or $k=-6$ |
Thanks for participating, kaliprasad and your answer is correct as well!1) $f1(x) and f2(x)$ must have a common factor. Otherwise g(x) shall be product of $ f1(x) and f2(x)$ and order 4
let $f1(x) = (x-m)(x-p)$
and let $f2(x) = 2(x-m)(x-q)$
comparing constant term
of f1(x) = mp = - a and 2 mq = a we get p = - 2q or m = 0 => a = 0
then taking the product and comparing coefficient of x
we get m+p = 29-a ...(1)
m – q = (43-2a)/2
or 2m – p = 43 – 2a ... (2)
solving (1) and (2) 3 m = (72-3a) or m = 24 – a
so p = 2m + 2a – 43 = 48 – 43 = 5
now - a = mp = 5(24-a) or 4a = 120 or a= 30 hence a = 30 is largest
f_1 | $f_1$ |