Factor:

$\displaystyle x^3-6x^2+9x$

Step 1: Factor out $\displaystyle x$

$\displaystyle x(x^2-6x+9)$

Step 2: Factor the polynomial

$\displaystyle x(x-3)(x-3)$

Factor:

$\displaystyle x^2+5x-14$

When factoring a polynomial $\displaystyle ax^2+bx+c$, the product of the coefficients must be $\displaystyle a$, the sum of the factors must be $\displaystyle b$, and the product of the factors must be $\displaystyle c$.

For the above equation, $\displaystyle a=1$, $\displaystyle b=5$, and $\displaystyle c=-14$.

Set up the factor equation:

$\displaystyle (x+{\color{Red} ?})(x+{\color{Red} ?})$

Becauase $\displaystyle c$ is negative, one of the factors must be negative as well.  Because $\displaystyle b$ is positive, this means the larger factor is positive as well.

Two numbers that meet these requirements are $\displaystyle -2$ and $\displaystyle 7$.  Their product is $\displaystyle -14$, and their sum is $\displaystyle 5$.

$\displaystyle (x-2)(x+7)$

Follow the FOIL rule when multiplying - first, outside, inside, last.

First: $\displaystyle x\cdot x=x^2$

Oustide: $\displaystyle x\cdot -9=-9x$

Inside: $\displaystyle 3\cdot x=3x$

Last: $\displaystyle 3\cdot -9=-27$

Add all of these together:

$\displaystyle x^2-9x+3x-27$

Combine like terms:

$\displaystyle {\color{Red} x^2-6x-27}$

Subtract 12 from both sides:

$\displaystyle x^2+x-12=0$

Factor:

$\displaystyle (x+3)(x-4)=0$

Then, by the zero-product property,

$\displaystyle x+3=0$ or $\displaystyle x-4=0$.

Solve:

$\displaystyle x=-3$ or $\displaystyle x=4$

Factor: $\displaystyle x^2+7x+12$

The last term is positive, which indicates that both factors should be positive:

$\displaystyle (x{\color{Red} +}?)(x{\color{Red} +}?)$

The sum of the factors must be $\displaystyle 7$, and their product must be $\displaystyle 12$.

This means the factors are $\displaystyle 3$ and $\displaystyle 4$

Therefore:

$\displaystyle {\color{Red} (x+3)(x+4)}$

When factoring trinomials with larger coefficients, rather than guess and check, there is a 4 step process that will always yield a correct answer.

Step 1 - Multiply the first and third terms

$\displaystyle 4x^{2}(-3)=-12x^{2}$

Step 2 - Find two factors that multiply to equal the result of step 1 ($\displaystyle -12x^{2}$) and add up to equal the second/middle term.

The product $\displaystyle -6x$ and $\displaystyle 2x=-12x^{2}$

Their sum is $\displaystyle -4x$     This works!!

So, $\displaystyle -6x$ and $\displaystyle 2x$ are our new factors

Step 3 - Replace our middle term ($\displaystyle -4x$) with our two new factors.

$\displaystyle 4x^{2}+2x-6x-3$

Step 4 - Factor by grouping to find your answer

$\displaystyle (4x^{2}+2x)-(6x+3)$

$\displaystyle 2x(2x+1) - 3(2x+1)$

$\displaystyle (2x-3)(2x+1)$   Answer!!!

To check your work, FOIL, and see if you get our original trinomial 

This can be solved by looking at all of the answers and multiplying them and comparing to the answer. However this is time consuming. You can start by noting that the term $\displaystyle -12$ can be a result of $\displaystyle 1\times12, 2\times 6, \text{ or }3\times 4$, where one of the terms is negative, so one answer can be eliminated. It is also clear that $\displaystyle 2x^{2}$ must be the result of $\displaystyle 2x$ multiplied by $\displaystyle x$, so two additional answers can be eliminated. Looking at the last two answers and multiplying through, the correct answer can be determined.

$\displaystyle (2x-3)(x+4)=0$

The polynomial factors into (x + 3) (3x - 2).

3x² + 7x – 6 = (a + b)(c + d)

There must be a 3x term to get a 3x² term.

3x² + 7x – 6 = (3x + b)(x + d)

The other two numbers must multiply to –6 and add to +7 when one is multiplied by 3.

b * d = –6 and 3d + b = 7

b = –2 and d = 3

3x² + 7x – 6 = (3x – 2)(x + 3)

(x + 3) is the correct answer.

The product of the last two numbers should be 6, while the sum of the products of the inner and outer numbers should be 5x. Factors of six include 1 and 6, and 2 and 3. In this case, our sum is five so the correct choices are 2 and 3. Then, our factored expression is (x + 2)(x + 3). You can check your answer by using FOIL.

y = x² + 5x + 6

2 * 3 = 6 and 2 + 3 = 5

(x + 2)(x + 3) = x² + 5x + 6

You can set the two numbers to equal variables, so that you can set up the algebra in this problem. The first odd number can be defined as $\displaystyle x$ and the second odd number, since the two numbers are consecutive, will be $\displaystyle x+2$.

This allows you to set up the following equation to include the given product of 195:

$\displaystyle x(x+2)=195$

$\displaystyle x^2 + 2x = 195$

Next you can subtract 195 to the left and set the equation equal to 0, which allows you to solve for $\displaystyle x$:

$\displaystyle x^2+2x-195=0$

You can factor this quadratic equation by determining which factors of 195 add up to 2. Keep in mind they will need to have opposite signs to result in a product of negative 195:

$\displaystyle (x \ \ \ \ \ $(x \ \ \ \ \ \) = 0\)

$\displaystyle (x -13)(x +15) = 0$

Set each binomial equal to 0 and solve for $\displaystyle x$. For the purpose of this problem, you'll only make use of the positive value for $\displaystyle x$:

$\displaystyle x-13 = 0 \ \ \ \ \ \ x +15 = 0$

$\displaystyle x = 13$

Now that you have solved for $\displaystyle x$, you know the two consecutive odd numbers are 13 and 15. You solve for the answer by finding the sum of these two numbers:

$\displaystyle 13+15=28$