# Which has a larger root?

#### anemone

##### MHB POTW Director
Staff member
Which is larger, the real root of x + x2 + ... + x8 = 8 - 10x9, or the real root of x + x2 + ... + x10 = 8 - 10x11?

#### HallsofIvy

##### Well-known member
MHB Math Helper
The second one. The first has root about 0.882, the second, bout 0.884. I got that by graphing both and then "zooming" in on the zeros.

#### Jester

##### Well-known member
MHB Math Helper
Here's my solution
It's not hard to show that the derivative of each is positive so both are increasing function meaning there's only one root. Let

$f(x) = 1 + x + x^2+ \cdots +x^8 + 10x^9 - 8$

and

$g(x) = 1 + x + x^2+ \cdots + x^{10} + 10x^{11} - 8$

We also have $f(0.8) <0, \; f(0.9) >0, \;g(0.8) <0, \;g(0.9) >0$ meaning that both roots lie between $0.8$ and $0.9$. Now consider the difference $h(x) = f(x)-g(x) =9x^8-x^{10}-10x^{11}= -(x+1)(10x-9)x^9$. On the interval $(0.8,0.9)$ $h(x)> 0$ meaning that $f(x) > g(x)$ giving that the root of $g(x)$ is larger than the root of $f(x)$ as HallsofIvy pointed out.

#### anemone

##### MHB POTW Director
Staff member
The second one. The first has root about 0.882, the second, bout 0.884. I got that by graphing both and then "zooming" in on the zeros.
Here's my solution
It's not hard to show that the derivative of each is positive so both are increasing function meaning there's only one root. Let

$f(x) = 1 + x + x^2+ \cdots +x^8 + 10x^9 - 8$

and

$g(x) = 1 + x + x^2+ \cdots + x^{10} + 10x^{11} - 8$

We also have $f(0.8) <0, \; f(0.9) >0, \;g(0.8) <0, \;g(0.9) >0$ meaning that both roots lie between $0.8$ and $0.9$. Now consider the difference $h(x) = f(x)-g(x) =9x^8-x^{10}-10x^{11}= -(x+1)(10x-9)x^9$. On the interval $(0.8,0.9)$ $h(x)> 0$ meaning that $f(x) > g(x)$ giving that the root of $g(x)$ is larger than the root of $f(x)$ as HallsofIvy pointed out.
Thanks for participating to both of you!