### All Algebra 1 Resources

## Example Questions

### Example Question #11 : Variables

Factor completely:

**Possible Answers:**

The polynomial cannot be factored further.

**Correct answer:**

Rewrite this as

Use the -method by splitting the middle term into two terms, finding two integers whose sum is 1 and whose product is ; these integers are , so rewrite this trinomial as follows:

Now, use grouping to factor this:

### Example Question #12 : Variables

For what value of allows one to factor a perfect square trinomial out of the following equation:

**Possible Answers:**

**Correct answer:**

Factor out the 7:

Take the 8 from the x-term, cut it in half to get 4, then square it to get 16. Make this 16 equal to C/7:

Solve for C:

### Example Question #8 : Factoring Polynomials

Factor the trinomial .

**Possible Answers:**

**Correct answer:**

We can factor this trinomial using the FOIL method backwards. This method allows us to immediately infer that our answer will be two binomials, one of which begins with and the other of which begins with . This is the only way the binomials will multiply to give us .

The next part, however, is slightly more difficult. The last part of the trinomial is , which could only happen through the multiplication of 1 and 2; since the 2 is negative, the binomials must also have opposite signs.

Finally, we look at the trinomial's middle term. For the final product to be , the 1 must be multiplied with the and be negative, and the 2 must be multiplied with the and be positive. This would give us , or the that we are looking for.

In other words, our answer must be

to properly multiply out to the trinomial given in this question.

### Example Question #13 : Variables

Factor:

**Possible Answers:**

**Correct answer:**

One way to factor a trinomial like this one is to put the terms of the polynomial into a box/grid:

Notice that there are 4 boxes but only 3 terms. To fix this, we find two numbers that add to 22 [the middle coefficient] and multiply to -48 [the product of the first and last coefficients]. By examining the factors of -48, we discover that these numbers must be -2 and 24.

Now we can put the terms into the box, by splitting the 22x into -2x and 24x:

To finish factoring, determine the greatest common factor of each of the rows and columns. For instance, and have a greatest common factor of .

Our final answer just combines the factors on the top and side into binomials. In this case, .

### Example Question #14 : Variables

Factor:

**Possible Answers:**

**Correct answer:**

One way to factor a trinomial like this one is to put the terms of the polynomial into a box/grid:

Notice that there are 4 boxes but only 3 terms. To fix this, we find two numbers that add to -7 [the middle coefficient] and multiply to -30 [the product of the first and last coefficients]. By examining the factors of -30, we discover that these numbers must be -10 and 3.

Now we can put the terms into the box, by splitting the -7x into -10x and 3x:

To finish factoring, determine the greatest common factor of each of the rows and columns. For instance, and have a greatest common factor of , and and -15 have a greatest common factor of 3.

Our final answer just combines the factors on the top and side into binomials. In this case, .

### Example Question #13 : Variables

Factor the trinomial:

**Possible Answers:**

**Correct answer:**

The first step in setting up our binomial is to figure out our possible leading terms. In this case, the only reasonable factors of are :

Next, find the factors of . In this case, we could have or . Either combination can potentially produce , so the signage is important here.

Note that since the last term in the ordered trinomial is positive, both factors must have the same sign. Further, since the middle term in the ordered triniomial is negative, we know the signs must both be negative.

Therefore, we have two possibilities, or .

Let's solve for both, and check against the original triniomial.

Thus, our factors are and .

### Example Question #16 : How To Factor A Trinomial

Factor the trinomial:

**Possible Answers:**

**Correct answer:**

The first step in setting up our binomial is to figure out our possible leading terms. In this case, the only reasonable factors of are :

Next, find the factors of . In this case, we could have or . Since the factor on our leading term is , and no additive combination of and can create , we know that our factors must be .

Note that since the last term in the ordered trinomial is negative, the factors must have different signs.

Therefore, we have two possibilities, or .

Let's solve for both, and check against the original triniomial.

Thus, our factors are and .

### Example Question #17 : How To Factor A Trinomial

Factor the trinomial:

**Possible Answers:**

**Correct answer:**

The first step in setting up our binomial is to figure out our possible leading terms. In this case, the only reasonable factors of are :

Next, find the factors of . In this case, we could have or . Since the factor on our leading term is , and no additive combination of and can create , we know that our factors must be .

Note that since the last term in the ordered trinomial is positive, the factors must have different signs. Since the middle term is also positive, the signs must both be positive.

Therefore, we have only one possibility, .

Let's solve, and check against the original triniomial.

Thus, our factors are and .

### Example Question #18 : How To Factor A Trinomial

Factor this trinomial, then solve for :

**Possible Answers:**

and

and

and

and

**Correct answer:**

and

Next, find the factors of . In this case, we could have or . Since the factor on our leading term is , and no additive combination of and or of and can create our middle term of , we know that our factors must be .

Note that since the last term in the ordered trinomial is positive, the factors must have the same sign. Since the middle term is negative, both factors must have a negative sign.

Therefore, we have one possibility, .

Let's solve, and check against the original triniomial, before solving for .

Thus, our factors are and .

Now, let's solve for . Simply plug and play:

Therefore, and .

Note that, in algebra, we can represent this by showing . However, writing the word "and" is perfectly acceptable.

### Example Question #16 : How To Factor A Trinomial

Factor the trinomial below,

**Possible Answers:**

**Correct answer:**

A factored trinomial is in the form , where the second term and the third term.

To factor a trinomial you first need to find the factors of the third term. In this case the third term is .

Factors of are:

The factors you choose not only must multiply to equal the third term, they must also add together to equal the second term.

In this case they must equal .

To check your answer substitute the factors of into the binomials and use FOIL.

First terms:

Outside terms:

Inside terms:

Last terms:

Simplify from here by combining like terms

Using FOIL returned it to the original trinomial, therefore the answer is: