# ISEE Upper Level Quantitative : Variables

## Example Questions

Simplify:

Explanation:

### Example Question #29 : Variables And Exponents

Expand:

Explanation:

A binomial can be cubed using the pattern:

Set

### Example Question #30 : Variables And Exponents

Factor completely:

Explanation:

A trinomial whose leading term has a coefficent other than 1 can be factored using the -method. We split the middle term using two numbers whose product is  and whose sum is . These numbers are , so:

### Example Question #81 : Variables

Multiply:

Explanation:

This can be achieved by using the pattern of difference of squares:

Applying the binomial square pattern:

### Example Question #251 : Algebraic Concepts

Simplify:

Explanation:

The cube of a sum pattern can be applied here:

### Example Question #252 : Algebraic Concepts

Fill in the box to form a perfect square trinomial:

Explanation:

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is , by 2, and square the quotient. The result is

### Example Question #1 : How To Multiply Exponential Variables

Fill in the box to form a perfect square trinomial:

Explanation:

To obtain the constant term of a perfect square trinomial, divide the linear coefficient, which here is , by 2, and square the quotient. The result is

### Example Question #251 : Algebraic Concepts

Expand:

Which is the greater quantity?

(a) The coefficient of

(b) The coefficient of

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

Explanation:

Using the Binomial Theorem, if  is expanded, the  term is

.

This makes  the coefficient of .

We compare the values of this expression at  for both  and :

(a)

(b)

(a) is the greater quantity.

### Example Question #254 : Algebraic Concepts

Which is the greater quantity?

(a)

(b) 8

It is impossible to tell from the information given

(a) and (b) are equal

(a) is greater

(b) is greater

(a) is greater

Explanation:

Since

, so

making (a) greater.

### Example Question #7 : How To Multiply Exponential Variables

Which is the greater quantity?

(a)

(b)

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.