# SAT II Math I : Simplifying Expressions

## Example Questions

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### Example Question #1 : How To Divide Monomial Quotients

Simplify the expression.

Explanation:

Because we are only multiplying terms in the numerator, we can disregard the parentheses.

To combine like terms in the numerator, we add their exponents.

To combine like terms between the numerator and denominator, subtract the denominator exponent from the numerator exponent.

Remember that any negative exponents stay in the denominator.

### Example Question #1 : Simplifying Expressions

Give the value of  that makes the polynomial  the square of a linear binomial.

None of the other responses gives a correct answer.

Explanation:

A quadratic trinomial is a perfect square if and only if takes the form

for some values of  and .

, so

and

For  to be a perfect square, it must hold that

,

so . This is the correct choice.

### Example Question #2 : Simplifying Expressions

Factor:

The polynomial is prime.

Explanation:

This can be factored out as the cube of a difference, where :

Therefore,

### Example Question #3 : Simplifying Expressions

Which of the following linear binomials is a factor of the polynomial ?

Explanation:

By the factor theorem, a polynomial  is divisible by the linear binomial  if and only if . We can use this fact to test each of the binomials by evaluating the dividend for the appropriate value of .

: Evaluate the polynomial at :

: Evaluate the polynomial at :

: Evaluate the polynomial at :

: Evaluate the polynomial at :

: Evaluate the polynomial at :

The dividend assumes the value of 0 at , so of the choices given,  is the factor.

### Example Question #1 : Simplifying Expressions

What is  increased by 40%?

Explanation:

A number increased by 40% is equivalent to 100% of the number plus 40% of the number. This is taking 140% of the number, or, equivalently, multiplying it by 1.4.

Therefore,  increased by 40% is 1.4 times this, or

### Example Question #2 : Simplifying Expressions

Which of the following is a prime factor of  ?

Explanation:

is the difference of two squares:

As such, it can be factored as follows:

The first factor is the sum of cubes and the second is the difference of cubes; each can be factored further:

Therefore,

Of the choices,  appears in the prime factorization and is therefore the correct choice.

### Example Question #1 : Simplifying Expressions

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

Explanation:

A number decreased by 20% is equivalent to 100% of the number minus 20% of the number. This is taking 80% of the number, or, equivalently, multiplying it by 0.8.

Therefore,  decreased by 20% is 0.8 times this, or

Divide:

Explanation:

Divide termwise:

### Example Question #111 : Single Variable Algebra

Simplify the expression:

Explanation:

To solve this problem, we first need to factor the numerator. We are looking for two numbers that multiply to equal -8 and sum to equal 2.

Now, we can write out our expression in fraction form.

Since we have the like term  in the numerator and denominator, we can cancel them out of our expression.

### Example Question #1 : Simplifying Expressions

Simplify:

Explanation:

To simplify, we begin by simplifying the numerator. When muliplying like bases with different exponents, their exponents are added.

For x:

For y:

For z:

The numerator is now .

When dividing like bases, their exponents are subtracted.

For x:

For y:

For z: