SAT Math : Variables

Study concepts, example questions & explanations for SAT Math

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Example Questions

Example Question #11 : Polynomials

F(x) = x^{3} + x^{2} - x + 2 

and

G(x) = x^{2} + 5  

What is ?

Possible Answers:

(FG)(x) = x^{3} + 2x^{2} - x + 7

(FG)(x) = x^{5} + x^{4} - x - 2

(FG)(x) = x^{5} + x^{4} - x^{3} + 2x^{2} - 5x -10

(FG)(x) = x^{3} - x - 3

(FG)(x) = x^{5} + x^{4} +4x^{3} + 7x^{2} - 5x +10

Correct answer:

(FG)(x) = x^{5} + x^{4} +4x^{3} + 7x^{2} - 5x +10

Explanation:

(FG)(x) = F(x)G(x) so we multiply the two function to get the answer.  We use x^{m}x^{n} = x^{m+n}

Example Question #11 : Polynomials

Find the product:

 

Possible Answers:

Correct answer:

Explanation:

Find the product:

Step 1: Use the distributive property.

Step 2: Combine like terms.

Example Question #11 : Polynomials

 represents a positive quantity;  represents a negative quantity.

Evaluate 

Possible Answers:

The correct answer is not among the other choices.

Correct answer:

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 #12 : Polynomials

 represents a positive quantity;  represents a negative quantity.

Evaluate .

Possible Answers:

Correct answer:

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 #11 : Polynomials

 and  represent positive quantities.

Evaluate .

Possible Answers:

Correct answer:

Explanation:

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

 

Additionally, 

 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 #11 : Polynomials

Simplify the following expression:

Possible Answers:

Correct answer:

Explanation:

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

Add like terms together:

has no like terms.

Combine these terms into one expression to find the answer:

Example Question #31 : Variables

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

For all real ,

How else could this operation be defined?

Possible Answers:

Correct answer:

Explanation:

, as the cube of a binomial, can be rewritten using the following pattern:

Applying the rules of exponents to simplify this:

Therefore, the correct choice is that, alternatively stated,

.

Example Question #21 : Polynomials

Solve for .

Possible Answers:

Correct answer:

Explanation:

Factor the expression

numerator: find two numbers that add to 2 and multiply to -8 [use 4,-2]

denominator: find two numbers that add to 5 and multiply to -14 [use 7,-2]

 

new expression:

Cancel the  and cross multiply.

Example Question #1 : Binomials

If 〖(x+y)〗= 144 and 〖(x-y)〗= 64, what is the value of xy?

 

 

Possible Answers:

16

20

22

18

Correct answer:

20

Explanation:

We first expand each binomial to get x2 + 2xy + y2 = 144  and x2 - 2xy + y2 = 64. We then subtract the second equation from the first to find 4xy = 80. Finally, we divide each side by 4 to find xy = 20.

 

 

Example Question #382 : Algebra

Solve each problem and decide which is the best of the choices given.

 

What are the zeros of the following trinomial?

Possible Answers:

Correct answer:

Explanation:

First factor out a . Then the factors of the remaining polynomial, 

, are  and .

Set everything equal to zero and you get , , and  because you cant forget to set  equal to zero.

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