# SAT II Math I : Single-Variable Algebra

## Example Questions

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### Example Question #121 : Single Variable Algebra

Factor completely:

Explanation:

The grouping technique works here:

The first factor is the difference of squares and can be factored further accordingly:

### Example Question #12 : Simplifying Expressions

Factor completely:

The polynomial is prime.

The polynomial is prime.

Explanation:

Since the first term is a perfect cube, the factoring pattern we are looking to take advantage of is the difference of cubes pattern. However, 243 is not a perfect cube of an integer , so the factoring pattern cannot be applied.  No other pattern fits, so the polynomial is a prime.

### Example Question #121 : Single Variable Algebra

Exponentiate:

Explanation:

Vertical multiplication is perhaps the easiest way to multiply trinomials.

### Example Question #123 : Single Variable Algebra

Exponentiate:

Explanation:

The difference of two terms can be cubed using the pattern

Where :

### Example Question #281 : Sat Subject Test In Math I

How many of the following are prime factors of  ?

I)

II)

III)

IV)

Four

Two

None

Three

One

Three

Explanation:

Factor  all the way to its prime factorization.

can be factored as the difference of two perfect square terms as follows:

is a factor, and, as the sum of squares, it is a prime.  is also a factor, but it is not a prime factor - it can be factored as the difference of two perfect square terms. We continue:

Therefore, of the given four choices, only  is not a factor, so the correct response is three.

### Example Question #1 : How To Find The Solution Of A Rational Equation With A Binomial Denominator

Simplify the expression.

Explanation:

First, factor the numerator. We need factors that multiply to and add to .

We can plug the factored terms into the original expression.

Note that appears in both the numerator and the denominator. This allows us to cancel the terms.

### Example Question #11 : Simplifying Expressions

Simplify the following expression:

Explanation:

When simplifying an equation,you must find a common factor for all values in the equation, including both sides.

and,  can all be divided by  so divide them all at once

.

This leaves you with

.

### Example Question #122 : Single Variable Algebra

Simplify the expression

Explanation:

Simplify the numerator by multiplying in the  term

Cancel out like terms in the numerator and denominator.

### Example Question #123 : Single Variable Algebra

Simplify:

Explanation:

Rewrite the denominator of the second fraction using a power.

Using the subtraction rule of exponents, we can simplify this as one term.

The expression becomes:

Apply the addition rule of exponents.