\(\displaystyle x^2 -10x - 24\)

To factor, we are looking for two terms that multiply to give \(\displaystyle -24\) and add together to get \(\displaystyle -10\).

Possible factors of \(\displaystyle -24\):

\(\displaystyle (-1,24),\ (1,-24),\ (-2,12),\ (2,-12),\ (-3,8),\ (3,-8),\ (-4,6),\ (4,-6)\)

Based on these options, it is clear our factors are \(\displaystyle 2\) and \(\displaystyle -12\).

\(\displaystyle (2)(-12)=-24\ \text{and}\ 2+(-12)=-10\)

Our final answer will be:

\(\displaystyle (x-12)(x+2)\)

\(\displaystyle x^2 +7x - 8\)

To factor, we are looking for two terms that multiply to give \(\displaystyle -8\) and add together to get \(\displaystyle 7\).

Possible factors of \(\displaystyle -8\):

\(\displaystyle (-1,8),\ (1,-8),\ (-2,4),\ (2,-4)\)

Based on these options, it is clear our factors are \(\displaystyle -1\) and \(\displaystyle 8\).

\(\displaystyle (-1)(8)=-8\ \text{and}\ (-1)+8=7\)

Our final answer will be:

\(\displaystyle (x+8)(x-1)\)

\(\displaystyle x^2 - 11x + 10\)

To factor, we are looking for two terms that multiply to give \(\displaystyle 10\) and add together to get \(\displaystyle -11\).

Possible factors of \(\displaystyle 10\):

\(\displaystyle (1,10),\ (-1,-10),\ (2,5),\ (-2,-5)\)

Based on these options, it is clear our factors are \(\displaystyle -1\) and \(\displaystyle -10\).

\(\displaystyle (-1)(-10)=10\ \text{and}\ (-1)+(-10)=-11\)

Our final answer will be:

\(\displaystyle (x-10)(x-1)\)

\(\displaystyle x^2 + 20x + 96\)

To factor, we are looking for two terms that multiply to give \(\displaystyle 96\) and add together to get \(\displaystyle 20\). There are numerous factors of \(\displaystyle 96\), so we will only list a few.

Possible factors of \(\displaystyle 96\):

\(\displaystyle (3,32),\ (-3,-32),\ (4,24),\ (-4,-24),\ (6,16),\ (8,12)\)

Based on these options, it is clear our factors are \(\displaystyle 8\) and \(\displaystyle 12\).

\(\displaystyle (8)(12)=96\ \text{and}\ 8+12=20\)

Our final answer will be:

\(\displaystyle (x + 12)(x+8)\)

\(\displaystyle x^2 -38x -600\)

To factor, we are looking for two terms that multiply to give \(\displaystyle -600\) and add together to get \(\displaystyle -38\). There are numerous factors of \(\displaystyle -600\), so we will only list a few.

Possible factors of \(\displaystyle -600\):

\(\displaystyle (2,-300),\ (3,-200),\ (5,-120),\ (6,-100),\ (10,-60),\ (12,-50),\ (15,-40)\)

Based on these options, it is clear our factors are \(\displaystyle 12\) and \(\displaystyle -50\).

\(\displaystyle (12)(-50)=-600\ \text{and}\ 12+(-50)=-38\)

Our final answer will be:

\(\displaystyle (x-50)(x+12)\)

\(\displaystyle x^2 - 26x + 105\)

To factor, we are looking for two terms that multiply to give \(\displaystyle 105\) and add together to get \(\displaystyle -26\).

Possible factors of \(\displaystyle 105\):

\(\displaystyle (3,35),\ (-3,-35),\ (5,21),\ (-5,-21),\ (7,15),\ (-7,-15)\)

Based on these options, it is clear our factors are \(\displaystyle -5\) and \(\displaystyle -21\).

\(\displaystyle (-5)(-21)=105\ \text{and}\ (-5)+(-21)=-26\)

Our final answer will be:

\(\displaystyle (x - 5)(x-21)\)

Factor the numerator:

\(\displaystyle x^2+6x+9\rightarrow (x+3)(x+3)\)

Simplify the fraction:

\(\displaystyle \frac{(x+3)(x+3)}{(x+3)}\rightarrow (x+3)\)

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:

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