GMAT Math : Understanding factoring

Example Questions

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Example Question #1 : Understanding Factoring

Factor .

Explanation:

To factor this, we need two numbers that multiply to  and sum to . The numbers  and  work.

Example Question #2 : Understanding Factoring

Factor .

Explanation:

is a difference of squares. The difference of squares formula is

So .

Then, .

Example Question #3 : Understanding Factoring

Solve .

Explanation:

Let's factor the expression: .

We need to look at the behavior of the function to the left and right of 1 and 5.  To the left of ,

You can check this by plugging in any value smaller than 1. For example, if ,

,

which is greater than 0.

When  takes values in between 1 and 5, .  Again we can check this by plugging in a number between 1 and 5.

, which is less than 0, so no numbers between 1 and 5 satisfy the inequality.

When  takes values greater than 5,

To check, let's try .  Then:

so numbers greater than 5 also satisfy the inequality.

Therefore .

Example Question #4 : Understanding Factoring

Solve .

Explanation:

First let's factor:

x < -8: Let's try -10.  (-10 + 8)(-10 - 1) = 22, so values less than -8 don't satisfy the inequality.

-8 < x < 1: Let's try 0.  (0 + 8)(0 - 1) = -8, so values in between -8 and 1 satisfy the inequality.

x > 1: Let's try 2.  (2 + 8)(2 - 1) = 10, so values greater than 1 don't satisfy the inequality.

Therefore the answer is -8 < x < 1.

Example Question #5 : Understanding Factoring

Factor the expression completely:

Explanation:

This expression can be rewritten:

As the difference of squares, this can be factored as follows:

As the sum of squares with relatively prime terms, the first factor is a prime polynomial. The second factor can be rewritten as the difference of two squares and factored:

Similarly, the middle polynomial is prime; the third factor can be rewritten as the difference of two squares and factored:

This is as far was we can factor, so this is the complete factorization.

Example Question #6 : Understanding Factoring

Where does this function cross the -axis?

It never crosses the x axis.

Explanation:

Factor the equation and set it equal to zero.  . So the funtion will cross the -axis when

Example Question #7 : Understanding Factoring

If , and , what is the value of ?

Explanation:

This questions tests the formula: .

Therefore, we have . So

Example Question #8 : Understanding Factoring

Factor:

Explanation:

can be grouped as follows:

is a perfect square trinomial, since

Now use the difference of squares pattern:

Example Question #9 : Understanding Factoring

Factor completely:

Explanation:

Group the first three terms and the last three terms, then factor out a GCF from each grouping:

We try to factor  as a sum of cubes; however, 5 is not a perfect cube, so the binomial is a prime.

To factor out , we try to factor it into , replacing the question marks with two integers whose product is 2 and whose sum is 3. These integers are 1 and 2, so

The original polynomial has  as its factorization.

Example Question #10 : Understanding Factoring

Factor completely:

Explanation:

Group the first three terms and the last three terms, then factor out a GCF from each grouping:

is the sum of cubes and can be factored using this pattern:

We try to factor out the quadratic trinomial as , replacing the question marks with integers whose product is 1 and whose sum is . These integers do not exist, so the trinomial is prime.

The factorization is therefore

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