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Basic Arithmetic : Factoring

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Example Questions

Example Question #1 : Factoring

Factor the following expression completely:

x^2+10x+24

Possible Answers:

(x+12)(x+2)

(x-4)(x-6)

(x+4)(x+6)

(x+3)(x+8)

Correct answer:

(x+4)(x+6)

Explanation:

To factor an expression in the form x^2+bx+c , we need to find factors of that add up to .

In this case, c=24 and b=10 .

Start by listing factors of 24 and adding them up. You want the one that adds up to 10.

1+24=25

2+12=14

3+8=11

4+6=10

Because 4 and 6 are the factors that we need, you can then write

x^2+10x+24=(x+4)(x+6)

To check if you factored correctly, you can multiply the two factors together. If you end up with the original expression, then you are correct.

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Example Question #2 : Factoring

Factor the following expression completely

$\frac{x^2-3x-54}{x^2+4x-12}$

Possible Answers:

$\frac{x-6}{x-2}$

$\frac{x-9}{x-2}$

$\frac{(x-9)(x-6)}{(x-2)}$

(x-9)(x-6)

Correct answer:

$\frac{x-9}{x-2}$

Explanation:

First, we need to factor the numerator and the denominator separately.

To factor an expression with the form x^2+bx+c , we will need to find factors of that add up to be.

For the numerator, x^2-3x-54 , c=-54 and b=-3 .

Write down the factors of and add them up.

1+(-54)=-53

2+(-27)=-25

3+(-18)=-15

6+(-9)=-3

Since and add up to , x^2-3x-54=(x-9)(x+6)

Now, do the same thing with the denominator, x^2+4x-12 .

-1+12=11

-2 + 6 = 4

Since and add up to , x^2+4x-12=(x+6)(x-2) .

Now, stack these factors up as fractions:

$\frac{x^2-3x-54}{x^2+4x-12}=\frac{(x-9)(x+6)}{(x+6)(x-2)}$

Since both the numerator and denominator have the factors (x+6), they cancel each other out because they divide to 1.

Then,

$\frac{(x-9)(x+6)}{(x+6)(x-2)}=\frac{x-9}{x-2}$

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Example Question #2 : Factoring

Factor the following expression: $\small \small x^3+14x^2+48x$

Possible Answers:

$\small x(x+7)^2$

$\small (x+2)(x+3)(x+8)$

$\small x(x^2+14x+48)$

$\small x(x+5)(x+9)$

$\small x(x+6)(x+8)$

Correct answer:

$\small x(x+6)(x+8)$

Explanation:

When you factor an expression, you are separating it into its basic parts. When you multiply those parts back together, you should obtain the original expression.

The first step when factoring an expression is to see if all of the terms have something in common. In this case, $\small x^3$ , $\small 14x^2$ , and $\small 48x$ all have an $\small x$ which can be taken out:

$\small x^3+14x^2+48x=x(x^2+14x+48)$

The next step is to focus on what's in the parentheses. To factor an expression of form $\small x^2+bx+c$ , we want to try to find factors $\small (x+m)(x+n)$ , where $\small m+n=b$ and $\small m\times n=c$ . We therefore need to look at the factors of $\small 48$ to see if we can find two that add to $\small 14$ :

$\small 1+48=49$

$\small 2+24=26$

$\small 3+16=19$

4+12=16

6+8=14

We've found our factors! We can therefore factor what's inside the parentheses, (x^2+14x+48) , as (x+6)(x+8) . If we remember the we factored out to begin with, our final completely factored answer is:

$\small x(x+6)(x+8)$

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Example Question #1 : Factoring

$7+66\div(12+7\cdot3)=?$

Possible Answers:

$\frac{73}{33}$

Correct answer:

Explanation:

This question requires you to understand order of operations, which is represented by the acronym "PEMDAS": parentheses, exponents, multiplication and division, addition and subtraction.

Solve the expression within the parenthesis first, beginning with multiplication:

$7+66\div(12+21)$

$7+66\div(33)$

The next operation in the order of operations is division.

7+2

Finally, use addition to solve the equation:

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Example Question #4 : Factoring

Factor x^2+12x+36

Possible Answers:

(x+9)(x+4)

(x+6)^2

(x+12)^2

(x+12)

(x+2)(x+3)

Correct answer:

(x+6)^2

Explanation:

To factor an equation in the form ax^2+bx+c , where b=12 and c=36 , you must find factors of that add up to .

List the factors of 36 and add them together:

1+36=37

9+4=13

12+3=15

6+6=12

Since 6+6=12 , is the factor we need. Plug this factor in to get the final answer.

x^2+12x+36 = (x+6)(x+6)= (x+6)^2

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Basic Arithmetic : Factoring

Study concepts, example questions & explanations for Basic Arithmetic

All Basic Arithmetic Resources

Example Questions

Example Question #1 : Factoring

Example Question #2 : Factoring

Example Question #2 : Factoring

Example Question #1 : Factoring

Example Question #4 : Factoring

All Basic Arithmetic Resources

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