MHB Find Largest a in Polynomial Factorization

[anemone]

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 $g(x)$?

 

[Opalg]

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 $g(x)$?

[sp]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 $a$.

Edit. The thing I was missing is the possibility that $a=0$. Then the common factor is $x$, so that $f_1(x) = x(x-29)$, $f_2(x) = x(2x-43)$, and $g(x) = x(x-29)(2x-43)$. So $a=30$ is in fact the "largest" value of $a$.[/sp]

 

[anemone]

[sp]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 $a$.[/sp]

Thanks for participating, Opalg!:)

Hmm, I can't see anything incomplete in your solution but I got two values of $a$, one is 0 and the other one is 30 and hence the largest $a$ would be 30.

Edit: I didn't realize you have already edited your post mentioned that you have found the missing case.

BTW, here is my solution:

Spoiler

Let $k$ be the common root of the two quadratic equations $f_1(x)=x^2+(a-29)x-a$ and $f_2(x)=2x^2+(2a-43)x+a$ to the polynomial $g(x)$.

By substituting $x=k$ into these two quadratic equations we get:

[TABLE="class: grid, width: 500"]
[TR]
[TD]$f_1(k)=k^2+(a-29)k-a=0$[/TD]
[TD]$f_2(k)=2k^2+(2a-43)k+a=0$[/TD]
[/TR]
[TR]
[TD]$k^2+(a-29)k-a=0$[/TD]
[TD]$2k^2+(2a-43)k+a=0$---(2)[/TD]
[/TR]
[TR]
[TD]$2k^2+2(a-29)k-2a=0$---(1)[/TD]
[TD][/TD]
[/TR]
[/TABLE]

Now, subtract equation (2) from equation (3) and solve for $a$, we get

$(2k^2+2(a-29)k-2a)-(2k^2+(2a-43)k+a)=0$

$a=-5k$

Back substitute $x=k$ and $a=-5k$ into the quadratic equation of $f_1$ and $f_2$ respectively gives

[TABLE="class: grid, width: 500"]
[TR]
[TD]$f_1(k)=k^2+(-5k-29)k+5k=0$[/TD]
[TD]$f_2(k)=2k^2+(2(-5k)-43)k-5k=0$[/TD]
[/TR]
[TR]
[TD]$k^2-5k^2-29k+5k=0$

$-4k^2-24k=0$

$k(k+6)=0$

$k=0$ or $k=-6$[/TD]
[TD]$2k^2-10k^2-43k-5k=0$

$-8k^2-48k=0$

$k(k+6)=0$

$k=0$ or $k=-6$[/TD]
[/TR]
[/TABLE]

So, the largest value of $a$ must go with the smallest negative value of $k$, i.e. $a_{\text{largest}}=-5(-6)=30$.

 

[kaliprasad]

Re: Find the largest "a"

Spoiler

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

 

[anemone]

Re: Find the largest "a"

Spoiler

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

Thanks for participating, kaliprasad and your answer is correct as well!

Note that you can make subscript 1 and 2 to distinguish the function of $f$ in latex as shown in the example below:

[TABLE="class: grid, width: 500"]
[TR]
[TD]f_1[/TD]
[TD]$f_1$[/TD]
[/TR]
[/TABLE]