can someone help me understand this?

Question

It says "determine whether each trinomial is a perfect square trinomial. If so, factor it.

1. m^2+16m+64
2.9s^2-6s+1

What does it mean when to determine whether each trinomial is a perfect square trinomial?

Answer 1

A perfect square trinomial can be written as (a + b)^2 with a and b rational numbers. Usually we restrict a and b further, to integers.

How to tell? If you expand out (a+b)^2 you get (a+b)(a+b) = a^2 + 2ab + b^2. (FOIL it, don't simply distribute the exponent.) See the pattern: a perfect square on each end, and twice the product of the two roots in the middle. Note that the middle term can be plus or minus, but the end term has to be plus. Let's look at the examples you gave..
m^2 + 16m + 64
m^2 is a perfect square, the square of m.
64 is a perfect squrae, the square of + or - 8
The middle term is twice the product of the two roots, that is, 2x8xm. So this fits the pattern of m^2 + 2x8xm + 8^2 so it is a perfect square and you can factor it like this:
First write a pair of parenthesis with the sign in between that you see in the first position, in this case, +
(..+..)
next write the two roots in the two positions on either side of the +
(m+8)
Finally write the exponent on the outside
(m+8)^2
To prove it, multiply it out (FOIL, don't just distributte the exponent)
(m+8)(m+8) = m^2 + 8m + 8m + 8^2 =
m^2 + 16m + 64
There you have it. Let's look at the next example now...
9s^2 - 6s + 1
9s^2 is a perfect square, because (3s)^2 = 9s^2
1 is a perfect square, becaues (-1)^2 = 1^2 = 1
In the middle you have -6s, wihch is twice the product of the two roots 2x(3s)x(-1) Note we're using the -1 this time, so as to get the negative sign in the second position.
Factor like before. First do this:
(..-..)
Then put the roots in
(3x - 1)
Finally put the exponent on
(3x - 1)^2
To prove it, you can multiply it out. There you go!

Answer 2

A perfect square is one that can be expressed like
(x - r)^2, or (x - r) (x - r).

For example, x^2 + 2x + 1 is a perfect square, because it can be expressed as (x + 1) (x + 1) = (x + 1)^2

1. m^2 + 16m + 64

This is a perfect square, because it factors into
(m + 8)^2

2. 9s^2 - 6s + 1

This is a perfect square, because it factors into
(3s - 1)^2

Answer 3

1. m^2+16m+64
64m^2=8m *8m
m^2+8m+8m+64
You make 2 groups
m^2+8m_______8m+64
m(m+8)_______8(m+8)
(m+8) (m+8)
It is a perfect square trinomial.

2.9s^2-6s+1
9s^2=-3*-3
9s^2-3-3+1
You group them
9s^2-3s_______-3s+1
3s(3s-1)_____ -1(3s-1)
(3s-1) (3s-1)
It is a perfect square trinomial