# SAT Math : Polynomials

## Example Questions

### Example Question #2 : Multiplying And Dividing Polynomials

Simplify:

Explanation:

Cancel by subtracting the exponents of like terms:

### Example Question #12 : Polynomial Operations

Divide  by .

Explanation:

It is not necessary to work a long division if you recognize  as the sum of two perfect cube expressions:

A sum of cubes can be factored according to the pattern

,

so, setting ,

Therefore,

### Example Question #13 : Polynomial Operations

By what expression can  be multiplied to yield the product ?

Explanation:

Divide  by  by setting up a long division.

Divide the lead term of the dividend, , by that of the divisor, ; the result is

Enter that as the first term of the quotient. Multiply this by the divisor:

Subtract this from the dividend. This is shown in the figure below.

Repeat the process with the new difference:

Repeating:

The quotient - and the correct response - is .

### Example Question #14 : Polynomial Operations

and

What is ?

Explanation:

so we multiply the two function to get the answer.  We use

### Example Question #15 : Polynomial Operations

Find the product:

Explanation:

Find the product:

Step 1: Use the distributive property.

Step 2: Combine like terms.

### Example Question #16 : Polynomial Operations

represents a positive quantity;  represents a negative quantity.

Evaluate

The correct answer is not among the other choices.

Explanation:

The first two binomials are the difference and the sum of the same two expressions, which, when multiplied, yield the difference of their squares:

Again, a sum is multiplied by a difference to yield a difference of squares, which by the Power of a Power Property, is equal to:

, so by the Power of a Power Property,

Also, , so we can now substitute accordingly:

Note that the signs of  and  are actually irrelevant to the problem.

### Example Question #17 : Polynomial Operations

represents a positive quantity;  represents a negative quantity.

Evaluate .

Explanation:

can be recognized as the pattern conforming to that of the difference of two perfect cubes:

Additionally, by way of the Power of a Power Property,

, making  a square root of , or 625; since  is positive, so is , so

.

Similarly,  is a square root of , or 64; since  is negative, so is  (as an odd power of a negative number is negative), so

.

Therefore, substituting:

.

### Example Question #18 : Polynomial Operations

and  represent positive quantities.

Evaluate .

Explanation:

can be recognized as the pattern conforming to that of the difference of two perfect cubes:

and  is positive, so

Using the product of radicals property, we see that

and

and  is positive, so

,

and

Substituting for  and , then collecting the like radicals,

.

### Example Question #31 : Variables

Simplify the following expression:

Explanation:

This is not a FOIL problem, as we are adding rather than multiplying the terms in parentheses.

has no like terms.

Combine these terms into one expression to find the answer:

### Example Question #12 : Polynomials

Define an operation  on the set of real numbers as follows:

For all real ,

How else could this operation be defined?