Describe your strategy for factoring a polynomial. Give an example and show all steps?

Answer 1

Simple quadratics
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Factor x^2 + 7x + 10

This will be factored in the form
(x + ?) (x + ?)

Notice that 7 is the coefficient of x, and 10 is the number with no coefficient. At this point, this is where you ask yourself: What two numbers multiply to make 10, and add to make 7?

The answer to that is 5 and 2, so 5 and 2 both go where the question marks are.

(x + 5) (x + 2)

Another example: x^2 - x + 42.

What two numbers multiply together to make 42, and add together to make -1? Here are the ways to get 42:
2 x -21, -2 x 21
-3 x 14, 3 x -14
-6 x 7, 6 x -7

The answer would be 6 and -7. Therefore,

x^2 - x + 42 = (x + 6) (x - 7)

More sophisticated quadratics
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3x^2 + 2x - 8

In this case, notice there's a number as the coefficient of x^2. These ones aren't as simple.

You would split 3x^2 up like you would the x^2; this time though, it splits into 3x and x. So our factoring would look like

(3x + ?) (x + ?)

Look at our coefficientless number, -8. What we have to do is plug in factors of -8 into the question marks, and ensure that the OUTER product plus the INNER product is equal to the coefficient of x.

Let's test 1 and -8. (3x + 1) (x - 8)
If we do so, the product of the outer terms would be
(-8)(3x) + (1)(x) = -24x + x = -23x. This is NOT equal to our middle term.

How about -8 and 1? (3x - 8) (x + 1).
Outer = 3x, inner is -8x, added together is -5x. Nope; still no good.

Let's test -2 and 4. (3x - 2) (x + 4)
Outer = 12x, inner = -2x, Outer plus Inner is 10x. No good.

Test 4 and -2: (3x + 4) (x - 2).
Outer = -6x, Inner = 4x, Outer + inner = -2x. This is CLOSE; our goal was to obtain 2x, NOT -2x. Because only the sign is different, all we have to do is negate each number (making them into -4 and 2), and this should be our answer.

Test -4 and 2: (3x - 4) (x + 2)
Outer = 6x, inner = -4x, outer plus inner is 2x. This is CORRECT.

Therefore, our factorization is 3x^2 + 2x - 8 = (3x - 4) (x + 2)

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Sometimes, you would get an example like
4x^2 + 5x + 1

Where, you have a choice between factoring it as
(4x + ?) (x + ?)
OR
(2x + ?) (2x + ?)

The reason is because 4x^2 can be split up into 4x and x, or 2x and 2x. In this case, what you would do is trial and error; if you reach a dead end trying one set of factors, try the other option.

Answer 2

1.Factoring Polynomials

The first thing you should always do when factoring is to take out a common factor. This is the simplest technique of factoring, but it is important even when you learn fancier techniques, because you will make your later work much easier if you always look for common factors first. Taking out common factors is using the distributive property backwards. The distributive property says
a(b+c)=ab+ac.

The idea behind taking out a common factor is to look for something like the right side here where there is a common factor, here it would be a, and turn it into something like the left side to factor it.

Example:
4x5+12x4-8x3=4x3(x2+3x-8)
A good trick for finding the largest common factor when you are using this method to factor polynomials is to find the greatest common factor of the numbers and the smaller power of the variable, so here the greatest common factor of the numbers is 4 and the smallest power of x is 3, so we can take out 4x3 as a common factor.

2.Grouping
Grouping is a fancier technique that is based on taking out common factors. For grouping we split the polynomial in two pieces and take out common factors in each of them. If we get the same thing left over in each piece, then we can take that big thing out as a common factor, and this will factor the polynomial. It is, however, important to note that this only works if we do get the same thing. Simply taking out common factors of pieces of a polynomial is not factoring it. To factor a polynomial we must write the whole polynomial as a product of two polynomials.
Example:
5x3+10x2+3x+6=5x2(x+2)+3(x+2)=(5x2+3)(x+2)
Here we take out a common factor of 5x2 from the first two terms and a common factor of 3 from the second two terms. This would not be a good method of factoring except that this polynomial is kind of special in that what is left when taking out the common factors in both cases is x+2. Then what we do is take out x+2 as a common factor of the two big chunks. The idea is that anything we can do with x we can do with x+2, because it represents a number too. If it helps, blur the (x+2) or think of it as one big strange looking letter.

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Answer 3

read your book...

heres an example..

2x(squared)+5x+6
answer
(x+3) (x+2)

how?
find somethin that when u multiply it you'll get the 1st term
thats how i got "x"

then find somethin that when u multiply it will give you the last term
thats how i got "3 and 2"

and for the last condition when u add the numbers in step two you should get the 2nd term

gets?

heres another example

9x(squared)+36

here an easy one, if the polynomial your factoring is a perfect square just get the square root of the 1st and last term like so..

(3x+6) squared

thats just some of the example thats all i can remember we just finished this lesson about a week ago...