help with specific trinomials.

**Zellus** · April 25th 2009, 09:29 PM

right, so i know how to do trinomials in a normal case with only one pronumeral.

a^2 + 2a + 1
(a+1)(a+1)

and i can do non-monic trinomials no problem too, problem for me is when i discover a trinomial which has 2 pronumerals.

a^2 + 2ab + b^2
now i have no idea how i'm meant to work it out.
i know that this simplifies to (a+b)(a+b) because i can do the expansion for it.
but i have no idea how to simplify it and if it were slightly more complex
eg. k^2 + 8kp + p^2 i would not know what to do in the exam.

help please! could someone please show me how to work out these expressions a^2 + 2ab + b^2

thankyou very much in advance.

**mr fantastic** · April 25th 2009, 09:44 PM

Originally Posted by Zellus

right, so i know how to do trinomials in a normal case with only one pronumeral.

a^2 + 2a + 1
(a+1)(a+1)

and i can do non-monic trinomials no problem too, problem for me is when i discover a trinomial which has 2 pronumerals.

a^2 + 2ab + b^2
now i have no idea how i'm meant to work it out.
i know that this simplifies to (a+b)(a+b) because i can do the expansion for it.
but i have no idea how to simplify it and if it were slightly more complex
eg. k^2 + 8kp + p^2 i would not know what to do in the exam.

help please! could someone please show me how to work out these expressions a^2 + 2ab + b^2

thankyou very much in advance.

Complete the square:

$k^2 + 8p + p^2 = (k + 4p)^2 - 15p^2 = (k + 4p)^2 - (\sqrt{15} p)^2$ .

Now use the difference of two squares formula to get $k^2 + 8p + p^2 = (k - [\sqrt{15} - 4] p) (k + [ \sqrt{15} + 4] p)$ .

**Zellus** · April 25th 2009, 11:23 PM

Originally Posted by mr fantastic

Complete the square:

$k^2 + 8p + p^2 = (k + 4p)^2 - 15p^2 = (k + 4p)^2 - (\sqrt{15} p)^2$ .

how did you know that k^2 + 8p + p^2 when has it's square completed = (k + 4p)^2 - 15p^2 without having to go through a slow, laborious process of trial and error?

**mr fantastic** · April 26th 2009, 02:59 AM

Originally Posted by Zellus

how did you know that k^2 + 8p + p^2 when has it's square completed = (k + 4p)^2 - 15p^2 without having to go through a slow, laborious process of trial and error?

The key is to know that ${\color{red}A^2 + 2AB} + B^2 = (A + B)^2$ . So ${\color{red}k^2 + 8p} + 16p^2 = (k + 4p)^2$ .

Hence $k^2 + 8p + p^2 = k^2 + 8p + 16p^2 - 15p^2 = (k + 4p)^2 - 15p^2$ .

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