### All GMAT Math Resources

## Example Questions

### Example Question #1 : Solving By Factoring

Factor .

**Possible Answers:**

**Correct answer:**

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

### Example Question #2 : Solving By Factoring

Factor .

**Possible Answers:**

**Correct answer:**

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

So .

Then, .

### Example Question #3 : Solving By Factoring

Solve .

**Possible Answers:**

**Correct answer:**

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 : Solving By Factoring

Solve .

**Possible Answers:**

**Correct answer:**

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 : Solving By Factoring

Factor the expression completely:

**Possible Answers:**

**Correct answer:**

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 : Solving By Factoring

Where does this function cross the -axis?

**Possible Answers:**

It never crosses the x axis.

**Correct answer:**

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

### Example Question #7 : Solving By Factoring

If , and , what is the value of ?

**Possible Answers:**

**Correct answer:**

This questions tests the formula: .

Therefore, we have . So

### Example Question #8 : Solving By Factoring

Factor:

**Possible Answers:**

**Correct answer:**

can be grouped as follows:

is a perfect square trinomial, since

Now use the difference of squares pattern:

### Example Question #9 : Solving By Factoring

Factor completely:

**Possible Answers:**

**Correct answer:**

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 : Solving By Factoring

Factor completely:

**Possible Answers:**

**Correct answer:**

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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