GMAT Math : Understanding factoring

Study concepts, example questions & explanations for GMAT Math

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Example Questions

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Example Question #1 : Understanding Factoring

Factor 4y^{2}+4y-15.

Possible Answers:

\dpi{100} \small (2y-3)(2y+3)

\dpi{100} \small (2y-5)(2y+5)

\dpi{100} \small (2y-3)(2y+5)

\dpi{100} \small (y-3)(y+5)

\dpi{100} \small (2y+3)(2y-5)

Correct answer:

\dpi{100} \small (2y-3)(2y+5)

Explanation:

To factor this, we need two numbers that multiply to \dpi{100} \small 4\times -15=60 and sum to \dpi{100} \small 4. The numbers \dpi{100} \small -6 and \dpi{100} \small 10 work.

4y^{2}+4y-15 = 4y^{2} - 6y + 10y - 15 = 2y(2y-3) + 5(2y-3) = (2y-3)(2y+5)

Example Question #2 : Understanding Factoring

Factor \frac{y-x}{x^{2}-y^{2}}.

Possible Answers:

\frac{y-x}{x-y}

\frac{-1}{x+y}

\frac{y-x}{x+y}

\frac{1}{x+y}

\frac{1}{x^{2}-y^{2}}

Correct answer:

\frac{-1}{x+y}

Explanation:

{x^{2}-y^{2}} is a difference of squares. The difference of squares formula is

{a^{2}-b^{2}} = (a-b)(a+b)

So {x^{2}-y^{2}} = (x-y)(x+y).

Then, \frac{y-x}{x^{2}-y^{2}} = \frac{y-x}{(x+y)(x-y)} = \frac{-(x-y)}{(x+y)(x-y)} = \frac{-1}{x+y}.

Example Question #3 : Understanding Factoring

Solve x^{2}-6x+5>0.

Possible Answers:

Correct answer:

Explanation:

Let's factor the expression: x^{2}-6x+5 = (x-1)(x-5).

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

x^{2}-6x+5>0

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, x^{2}-6x+5 <0.  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, x^{2}-6x+5>0

To check, let's try .  Then:

so numbers greater than 5 also satisfy the inequality.

Therefore .

Example Question #4 : Understanding Factoring

Solve x^{2}+7x-8 <0.

Possible Answers:

Correct answer:

Explanation:

First let's factor: x^{2}+7x-8=(x+8)(x-1)

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:

Possible Answers:

Correct answer:

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?

Possible Answers:

It never crosses the x axis.

Correct answer:

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 ?

Possible Answers:

Correct answer:

Explanation:

This questions tests the formula: .

Therefore, we have . So

Example Question #8 : Understanding Factoring

Factor:

Possible Answers:

Correct answer:

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:

Possible Answers:

Correct answer:

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: 

Possible Answers:

Correct answer:

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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