Factoring Trinomials (When a = 1)?

Question

1: x2 - 8x + 16
(x - 8)(x - 2)
(x - 4)(x - 4)
(x - 4)(x + 4)


2: x2 + 8x - 48
(x + 12)(x - 4)
(x + 8)(x + 6)
(x + 8)(x - 6)


3: x2 - 6x - 16
(x - 4)(x - 4)
(x + 4)(x - 4)
(x - 8)(x + 2)


4: x2 - 2x - 48
(x - 12)(x + 4)
(x + 6)(x - 8)
(x - 6)(x - 8)


5: x2 + 14x + 33
(x + 11)(x + 3)
(x - 11)(x + 3)
(x + 33)(x + 1)


6: x2 - 2x + 1
(x - 1)(x + 1)
(x - 2)(x - 1)
(x - 1)(x - 1)


7: x2 - 5x + 4
(x - 1)(x - 4)
(x - 1)(x + 4)
(x - 2)(x - 2)


8: x2 - 10x - 11
(x - 11)(x - 1)
(x - 11)(x + 1)
(x + 11)(x + 1)


9: x2 - 11x + 30
(x - 5)(x - 6)
(x + 3)(x - 10)
(x - 3)(x - 10)


10: x2 - 9x - 10
(x - 5)(x + 2)
(x + 5)(x - 2)
(x + 1)(x - 10)


11: x2 - 4x - 5
(x + 1)(x - 5)
(x - 1)(x - 5)
(x - 1)(x +5)


12: x2 - 8x + 15
(x - 3)(x - 5)
(x - 15)(x - 1)
(x + 3)(x + 5)


13: x2 + 13x + 42
(x + 21)(x + 2)
(x + 2)(x - 21)
(x + 7)(x + 6)


14: 2x2 - 32x + 128
2(x - 16)(x - 4)
2(x - 8)(x - 8)
2(x - 2)(x + 32)


15: 2x2 + 4x - 198
2(x - 9)(x - 11)
2(x - 2)(x + 11)
2(x - 9)(x + 11)

Answer 1

x² - 8x + 16

x² - 4x - 4x + 16

x(x - 4) - 4(x - 4)

(x - 4(x - 4)

- - - - - - - - -

x² + 8x - 48

x² + 12x - 4x - 48

x(x + 12) - 4(x + 12)

)x - 4)(x + 12)

- - - - - - - - - - - - -

x² - 6x - 16

x² - 8x + 2x - 16

x(x - 8) + 2(x - 8)

(x + 2)(x - 8)

- - - - - - - - - - -s-

Answer 2

There are two ways to approach this problem.

If you just want answers, and don't care about factoring, or the Associative, Commutitive and Distributive properties, or don't care...
And if you are given choices from which to choose...
Try each of the choices until you find the right one.
1: x2 - 8x + 16
(x - 8)(x - 2)= x^2 - 2x -8x +16 = x^2 - 10x +16... Hmmm guess not
(x - 4)(x - 4)= x^2 - 4x - 4x +16 = x^2 - 8x + 16 BINGO
(x - 4)(x + 4) = x^2 +4x - 4x - 16.... Nope

But you may not always get choices from which to choose... of what if one of the choices is "none of the above"?
A "quadratic" is always of the form...
ax^2 + bx + c. [REF 1]
if it can be factored into (dx + e)(fx + g) [REF 2]
then a=df, c=eg, and b=dg+ef [REF 3]
Problem: x2 - 2x - 48
Choices:
(x - 12)(x + 4)
(x + 6)(x - 8)
(x - 6)(x - 8)

In this instance, a = 1 so df = 1 [REF 4]
In elementary math, this usually means that d and f are both 1, but this is not necessarily so. 2 multiplied by 1/2 also = 1 and there infinite other possibilities...
c = -48 so eg = -48... and we know that for eg to be negative, they must have differing signs. [REF 5 -- from REF 4]
To start, let's presume that we are looking at factors like
(x+e)(x+g)..we have already said one of e and g must be negative and the other positive {from REF 5]. And we also know from REF 3 that with d and f = 1, we have g + e = -2, and that eg = -48
1 x 48 = 2 x 24 = 3 x 16 = 4 x 12 = 6 x 8 = 48
Trying these pairs of factors... we could start with 1 and 48, but since the difference has to be -2, we know the factors have to be close together... 6 - 8 = -2 so we have

(x - 8)(x + 6)... now check
xx + 6x - 8x - 48 = x^2 - 2x - 48 BINGO...
Unfortunately you don't have (x - 8)(x + 6) as one of the choices.... but you do have (x + 6)(x - 8)... and the commutitive property for multiplication says
(x + 6)(x - 8)=(x - 8)(x + 6) that's the answer.

Now i could solve all the problems for you... but you're the one who's supposed to be learning...

Helpful hints...
First, try to get a = 1 by factoring a out of all the terms. If the terms get too weird, like 7/2 just realize that d and f may not both be 1. It makes the work a little less trivial... but still often "doable."

Second, if it gets too complicated and nasty.. just apply the quadratic equation. If you don't know it yet, you will soon. The quadratic formula is derived, by the way, using the very processes we've been discussing.

Third, not everything can be factored using the Real numbers..You should hear about Imaginary number soon too.

Answer 3

for number one: You can pull out the common factor of 2x^2 to get: 2x^2 (x^2-8x+15) To factor what is in paranthesis, we must first think: What two numbers multiplied together make 15 and added together make -8? After a little bit of thought, we see that -5*-3 is 15 and -5 plus -3 is -8, so our two factors are: (x-5)(x-3) SO the final factored answer is: 2x^2(x-5)(x-3) :) for the second problem, we can pull out a common factor of 7x to get: 7x(x^2 +7x+10) Now let us think what two numbers multiplied together make 10 and added together make 7? We come up with 5*2=10 and 5+2 = 7, so our two factors are: (x+5)(x+2) So the final answer would be: 7x(x+5)(x+2) :)

Answer 4

1)(x-4)(x-4) Two numbers that multiply to give +16 and add to give -8. Try the others for yourself if you get stuck..e-mail me..

Answer 5

My question was for you to explain a trinomial and the methoid to work it