# 代数学: 因数分解

1.
$$12{x}^{3}+11{x}^{2}+2x$$
解答
2.
$$60{h}^{2}+280h+45$$
解答
3.
$$8{x}^{3}-125$$
解答
4.
$$-3{x}^{2}+36x-108$$
解答
5.
$$3{x}^{3}+21{x}^{2}+36x$$
解答
6.
$$6{x}^{2}y+4xy+2ya$$
解答
7.
$${x}^{3}+5x+2{x}^{2}+10$$
解答

# Factoring - Introduction

A polynomial is an expression composed of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A common form of polynomials are quadratic expressions, which follows the form:
$$a{x}^{2}+bx+c$$
. For example:
$${x}^{2}+10x+16$$
Other variations might include extra terms, higher-power exponents or negative numbers. Regardless, in many cases, it is possible to turn the polynomial into a simpler form through a process called “factoring”. Let’s try a few simple factoring problems and find out how.

# The Simple Case

The simplest case is where there is no number in front of the
$${x}^{2}$$
, meaning that the cofficient is “1”. Let's use the previous expression as our example:
$${x}^{2}+10x+16$$
To factor this expression, we look for two numbers that multiply to
$$b$$
$$c$$
. In this case, these are
$$8$$
and
$$2$$
, which let us split the second term. This gives us:
$${x}^{2}+8x+2x+16$$
Next, we split the polynomial into two sets of terms, like this:
$$({x}^{2}+8x)+(2x+16)$$
Then we factor out the common terms to give:
$$x(x+8)+2(x+8)$$
Now, we factor out the common term again, which produces:
$$(x+2)(x+8)$$

# More Complex Cases

What happens if we are dealing with a more complex polynomial, such as:
$$12{x}^{3}+11{x}^{2}+2x$$
Here, we have a thrid-degree polynomial with a leading cofficient that is not
$$1$$
. Let's start by taking out the greatest common factor (GCF) of all three terms, which is
$$x$$
. This gives:
$$x(12{x}^{2}+11x+2)$$
Now, we want to do the same thing as above: Split the second term. Start by multiplying the coefficient of the first term,
$$12$$
, by the constant term,
$$2$$
. This gives
$$24$$
. Then, ask the same question as above — what numbers add up to
$$11$$
and multiply to
$$24$$
$$8$$
and
$$3$$
, which gives:
$$x(12{x}^{2}+8x+3x+2)$$
Now, we can split the polynomial in two and find the common term in
$$12{x}^{2} + 8x$$
, which is
$$4x$$
. This lets us remove the factor and gives:
$$x(4x(3x+2)+(3x+2))$$
Once we factor the common term
$$(3x+2)$$
, we arrive at the final answer:
$$x(3x+2)(4x+1)$$