# GED Math : Simplifying, Distributing, and Factoring

## Example Questions

### Example Question #31 : Algebra

Factor completely:

Explanation:

The polynomial is the difference of squares and can be factored using the pattern

where

as seen here:

### Example Question #32 : Simplifying, Distributing, And Factoring

Which of the following is a factor of the polynomial  ?

Explanation:

The greatest common factor of the two terms is the monomial term , so factor it out:

Of the four choices,  is correct.

### Example Question #31 : Simplifying, Distributing, And Factoring

Which of the following is a factor of the polynomial  ?

Explanation:

The greatest common factor of the two terms is the monomial term , so factor it out:

Of the four choices,  is correct.

### Example Question #34 : Simplifying, Distributing, And Factoring

Simplify:

Explanation:

Raise a fraction to a negative power by raising its reciprocal to the power of the absolute value of the exponent. Then apply the power of a quotient rule:

### Example Question #35 : Simplifying, Distributing, And Factoring

Simplify:

Explanation:

To raise a number to a negative exponent, raise it to the absolute value of that exponent, then take its reciprocal. We do this, then apply the various properties of exponents:

### Example Question #31 : Simplifying, Distributing, And Factoring

Factor completely:

Explanation:

For a quadratic trinomial with a quadratic coefficient other than 1, use the factoring by grouping method.

First, find two integers whose product is  (the product of the quadratic and constant coefficients) and whose sum is 1 (the implied coefficient of  ). By trial and error, we find that these are

Split the linear term accordingly, then factor by grouping, as follows.

### Example Question #35 : Simplifying, Distributing, And Factoring

Factor:

Explanation:

The greatest common factor of the terms is , so factor it out:

The trinomial might be able to be factored as

,

where  and .

By trial and error, we find that

,

so the factorization becomes

.

### Example Question #38 : Simplifying, Distributing, And Factoring

Decrease  by 40%. Which of the following will this be equal to?

Explanation:

A number decreased by 40% is equivalent to 100% of the number minus 40% of the number. This is taking 60% of the number, or, equivalently, multiplying it by 0.6.

Therefore,  decreased by 40% is 0.6 times this, or

.

### Example Question #31 : Simplifying, Distributing, And Factoring

Increase  by 20%. Which of the following will this be equal to?

Explanation:

A number increased by 20% is equivalent to 100% of the number plus 20% of the number. This is taking 120% of the number, or, equivalently, multiplying it by 1.2.

Therefore,  increased by 20% is 1.2 times this, or

.

### Example Question #40 : Simplifying, Distributing, And Factoring

Which of the following is a prime factor of ?

Explanation:

This can be most easily solved by first substituting  for , and, subsequently,  for :

This becomes quadratic in the new variable, and can be factored as

,

filling out the blanks with two numbers whose sum is  and whose product is . Through some trial and error, the numbers can be seen to be .

Therefore, after factoring and substituting back,

The first factor, the sum of squares, is prime. The second factors as the difference of squares, so the final factorization is

.

Of the choices given,  is correct.