### All SAT II Math I Resources

## Example Questions

### Example Question #1 : Simplifying Expressions

Simplify the expression.

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

None of the other responses gives a correct answer.

**Correct answer:**

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 #1 : Simplifying Expressions

Factor:

**Possible Answers:**

The polynomial is prime.

**Correct answer:**

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

Therefore,

### Example Question #1 : Simplifying Expressions

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

**Possible Answers:**

**Correct answer:**

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 #2 : Simplifying Expressions

What is increased by 40%?

**Possible Answers:**

**Correct answer:**

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 #1 : Simplifying Expressions

Which of the following is a prime factor of ?

**Possible Answers:**

**Correct answer:**

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 #3 : Simplifying Expressions

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

**Possible Answers:**

**Correct answer:**

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

### Example Question #2 : Simplifying Expressions

Divide:

**Possible Answers:**

**Correct answer:**

Divide termwise:

### Example Question #1 : Simplifying Expressions

Simplify the expression:

**Possible Answers:**

**Correct answer:**

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.

Thus, our answer is .

### Example Question #2 : Simplifying Expressions

Simplify:

**Possible Answers:**

**Correct answer:**

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:

Thus, our answer is .

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