### All ACT Math Resources

## Example Questions

### Example Question #1 : Binomials

What is the value of the coefficient in front of the term that includes in the expansion of ?

**Possible Answers:**

**Correct answer:**

Using the binomial theorem, the term containing the *x*^{2} *y*^{7 }will be equal to

(2*x*)^{2}(–*y*)^{7}

=36(–4*x*^{2} *y*^{7})= -144*x*^{2}*y*^{7}

### Example Question #2 : Binomials

A function of the form passes through the points and . What is the value of ?

**Possible Answers:**

**Correct answer:**

The easisest way to solve for is to begin by plugging each pair of coordinates into the function.

Using our first point, we will plug in for and for . This gives us the equation

.

Squaring 0 gives us 0, and multiplying this by still gives 0, leaving only on the right side, such that

.

We now know the value of , and we can use this to help us find . Substituting our second set of coordinates into the function, we get

which simplifies to

.

However, since we know , we can substitute to get

subtracting 7 from both sides gives

and dividing by 4 gives our answer

.

### Example Question #2 : Binomials

is equivalent to which of the following?

**Possible Answers:**

**Correct answer:**

To answer this problem, we need to multiply the expressions together, being mindful of how to correctly multiply like variables with exponents. To do this, we add the exponents together if the the like variables are being multiplied and subtract the exponents if the variables are being divided. So, for the presented data:

We then multiply the remaining expressions together. When we do this, we will multiply the coefficients together and combine the different variables into the final expression. Therefore:

This means our answer is .

### Example Question #3 : Binomials

Give the coefficient of in the product

.

**Possible Answers:**

**Correct answer:**

While this problem can be answered by multiplying the three binomials, it is not necessary. There are three ways to multiply one term from each binomial such that two terms and one constant are multiplied; find the three products and add them, as follows:

Add: .

The correct response is .

### Example Question #4 : Binomials

Give the coefficient of in the product

**Possible Answers:**

**Correct answer:**

While this problem can be answered by multiplying the three binomials, it is not necessary. There are three ways to multiply one term from each binomial such that two terms and one constant are multiplied; find the three products and add them, as follows:

Add:

The correct response is .

### Example Question #6 : Binomials

Give the coefficient of in the binomial expansion of .

**Possible Answers:**

**Correct answer:**

If the expression is expanded, then by the binomial theorem, the term is

or, equivalently, the coefficient of is

Therefore, the coefficient can be determined by setting

:

### Example Question #7 : Binomials

Give the coefficient of in the binomial expansion of .

**Possible Answers:**

**Correct answer:**

If the expression is expanded, then by the binomial theorem, the term is

or, equivalently, the coefficient of is

Therefore, the coefficient can be determined by setting

:

### Example Question #8 : Binomials

Give the coefficient of in the binomial expansion of .

**Possible Answers:**

**Correct answer:**

If the expression is expanded, then by the binomial theorem, the term is

or, equivalently, the coefficient of is

Therefore, the coefficient can be determined by setting

### Example Question #5 : Binomials

Give the coefficient of in the product

.

**Possible Answers:**

**Correct answer:**

While this problem can be answered by multiplying the three binomials, it is not necessary. There are three ways to multiply one term from each binomial such that two terms and one constant are multiplied; find the three products and add them, as follows:

Add:

The correct response is -122.

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