smart (in math) people wanted....?

Question

3. factor completely.

2a^4 - 18a^2

4. factor completely.

t^2 - (t - 1)^2

Answer 1

When factoring, you look for things that are common to all terms. It does not matter in what order you find them and it does not matter how many steps you take (as long as you don't get mixed up).

For example, in
2a^4 - 18a^2
we see that both coefficients (2 and 18) are even, so let's factor out '2'.

2(a^4 - 9a^2)
next, we see that both terms contain a^2 (because a^4 = a^2 times a^2); so we factor this out as well
2a^2(a^2 - 9)

We are left with a difference of squares for which there is a recipe (might as well learn it, you'll see it often):

x^2 - y^2 = (x+y)(x-y)
(This is the general recipe)
(let's apply it to our difference)

a^2 - 9 =
a^2 - 3^2 = (a+3)(a-3)

put it all together and you get your answer.

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For the second one, do it right away as a difference of squares, then factor each bracket separately

t^2 - (t-1)^2 =
[t+(t-1)][t-(t-1)]
(2t -1)(+1)
2t-1


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(This is a trick to quickly find squares mentally:

The difference between the square of a number and the square of the next number is the sum of the two numbers.

So if t= 85 and t-1=84 then the difference is 85+84 (same as 2 times 85, minus 1)

Someone asks you the square of 86?

Quickly find the square of 85:
it ends with 5 so its square will end in 25
the previous digits will be 8 times 9 (the digit times the next upper digit) = 72
square of 85 = 7225
to find square of 86, take 7225 and add 85+86 =
7225+85 = 7310, then 7310+86 = 7396.

Answer 2

3. 2a²(a²-9)=2a²(a-3)(a+3)
4. [t-(t-1)][t+(t-1)]