does it matter if you put it like this- (x+7)(-x+3)?

Question

I got (x+7)(-x+3) for the question "25-(2+x)^2"
Sometimes in equations, I get a - infront of the x like (-x+3). But the teacher's answers never have that minus. It always gets to (x-3)or something like that.
Does it really matter?Will i be wrong if i let my answer be like that?

Answer 1

It's not wrong, but most people DO prefer to begin their parenthetical expressions with a POSITIVE term, just as they prefer to start with the HIGHEST POWERS of ' x ' also (which you didn't do either --- see your original (2 + x)). [### --- see FOOTNOTE below.]

So, in line with these conventions, in cases like the one you've given, one puts the minus sign outside the rest of the expression. Thus your example would be written:

- (x + 7)(x - 3). [### --- see FOOTNOTE below.]

Following the usual conventions FROM THE START, you'd have

25 - (x + 2)^2 = 25 - x^2 - 4 x - 4 = - (x^2 + 4 x - 21)

= - (x + 7)(x - 3).

Your teacher should mark your answer as correct, but will probably suggest that your work would be a little more elegant if you followed the suggestions I've made. It's not only a matter of STYLE and AESTHETICS --- it has a very HELPFUL aspect to it, also. [ ### Once again, see my FOOTNOTE. ]

Live long and prosper.

### FOOTNOTE:

Note that there is a significant advantage for those distributively challenged in writing your expression for example as:

- (x + 7)(x - 3) rather than as (x + 7)(3 - x),

if the problem is to EXPAND rather than to produce such an expression.

SOME responders have endorsed writing the last parenthetical term as last written above, that is as (3 - x), but they are NOT good guides. This way of writing it has some significant disadvantages and inefficiencies.

Why is the first way of writing it superior, particularly if later expansion is needed? : Because then, the use of the well-known distributive mnemonic "FOIL" collects together, in the following order, (i) the x^2 term, (ii) and (iii) the x terms (thus easily combined, subsequently), and finally (iv) the constant term. Thus ALL FOUR TERMS are produced in the order that you WANT, for subsequent combination.

If, on the other hand, "FOIL" is applied to the second, alternative form, it produces successively (i) an x term, (ii) then the x^2 term, (iii) the constant term, and finally (iv) another x term. The trouble with this latter "mixed" way of writing it, then, is that "FOIL" gives you, as output, the different kinds of terms all mixed up, rather than in a more rational, and already logically well-arranged order. You then have to rearrange them before combining them together. This gives you many more opportunities to make a mistake.

Having a consistent ordering convention thus serves to lay out operations in a way that naturally speeds up calculations and also helps to prevent mistakes in gathering terms together. I strongly recommend it for its economy and efficiency.

Answer 2

That negative sign in front of the x does matter.
And yes, the correct answer has to have the negative sign.

The original given is 25-(2+x)^2 and if you expand this you get
21 -4x -x^2 , then factoring

you get (x+7)(-x+3) or (-x+3)(x + 7), either one is the correct answer.

However, if you do not want the minus sign inside the parenthesis, then factor out -1 from (-x +3) which gives you
-1(x-3). Putting it all together you can write an alternative answer as -1(x-3)(x+7) or just -(x-3)(x+7). And of course, the placement of the negative sign is very important and does matter. all the time.

The factored answer without the minus --- (x+7)(x+3) will be wrong because if you FOIL this you will get x^2 +10x +21, which is totally different from the given original question.

Answer 3

25 - (2 + x)^2
= [5 + (2 + x)][5 - (2 + x)]
= (5 + 2 + x)(5 - 2 - x)
= (x + 7)(-x + 3)

This is a correct answer, but most textbooks and standardized tests will factor out -1 so that the second factor has a leading coefficient of 1.

= (x + 7)[(-1)(x - 3)]
= (-1)(x + 7)(x - 3)
= -(x + 7)(x - 3)

The theory of linear factors like x - a use the assumption that the leading coefficient of the linear factor is 1. By writing linear factors in this manner makes it easier to discuss and interpret our calculations with this traditional form.

Answer 4

No, it does not matter. However, you might want to rewrite it as (3 - x). It's a little easier to understand (in my opinion) and it's an equivalent expression.

If you are unsure if an expression is equal to another one, plug in a test value for all the variables and evaluate. Be consistent with your values. For example, if you choose x=3 in one expression, choose x=3 for the other one.

So, when x=3,
25 - (2 + x)^2
25 - (2 + 3)^2
25 - 5^2
25 - 25
0

and,
(x + 7)(-x + 3)
(3 + 7)(-3 + 3)
(10)(0)
0

and,
(x + 7)(3 - x)
(3 + 7)(3 - 3)
(10)(0)
0

See? That is a quick and informal way to check your answers mentally if you are unsure. It's not a good way to solve a problem or to formally check it, just a little something you can do to boost your confidence.

Answer 5

For most factoring problems use my factoring calculator at http://www.poodwaddle.com/mathfactor.html

When the A term is negative it is best to factor out a common term of -1 For example:
-AX^2 + BX + C = 0
-1(AX^2 - BX - C) = 0
The negative is irrelevant because when moved to the other side of the equal sign (multiplicative inverse) it divides into 0 and disappears.

Concider this: To solve a factored problem each factor is made into an equation equaling zero so your factor of (-x+3) becomes:
-x+3=0
-x = -3
x = 3

Alternately if you swap the signs (x-3) your get:
x-3=0
x=3

If you ever run into a teacher that marks you wrong for this you should be able to prove to him that your solution results in the same values.

Answer 6

It is typical to not have a minus sign like that; to get rid of it just multiply the whole expression by -1 to get -(x+7)(x-3).

Answer 7

25-(2+x)^2 = (x+7)(-x+3)

You're right, if your teacher has (x+7)(x-3), he/she is wrong!

Please check if he/she wrote:

(x+7)(3-x) same result better style

Answer 8

Will not be wrong. It is mathematically correct. However, most people tend to start expressions of with the positive term as negative terms can easily be forgotten or confusing. If you understand what you are doing, then I won't bother correcting it as it may confuse you.