# GED Math : Solving by Factoring

## Example Questions

### Example Question #4603 : Algebra 1

Factor the polynomial.

Possible Answers:

Correct answer:

Explanation:

To factor a polynomial of the form , we want to look at the factors of  and the factors of . We want to find the combination of factors which when multiplied and added together give the value of .

In our case, , and .

The factors for  are .

The factors for  are

Since  is  we will want to use the factors  because .

Therefore when we put these factors into the binomal form we get,

.

Also see that

will foil out into the original polynomial, as , the coefficient for our  term, and , the constant.

### Example Question #11 : Solving By Factoring

Factor completely:

Possible Answers:

The expression is not factorable.

Correct answer:

Explanation:

First, factor out the greatest common factor (GCF), which here is . If you don't see the whole GCF at once, factor out what you do see (here, either the  or the ), and then check the result to see if any more factors can be pulled out.

Then, to factor a quadratic trinomial, list factors of the quadratic  term and the constant (no variables) term, then combine them into binomials that when multiplied back out will give the original trinomial.

Here, the quadratic term has only one factorization: .

The constant term has factorizations of , and .

We know the constant term is positive, so the binomials both have the same operation in them (adding or subtracting), since a positive times a positive OR a negative times a negative will both give a positive result.

But since the middle term  is negative, both binomial factors must contain subtraction. And .

### Example Question #5 : How To Factor A Variable

Solve for .

Possible Answers:

Correct answer:

Explanation:

Multiply both sides by 3:

Distribute:

Subtract  from both sides:

Add the  terms together, and subtract  from both sides:

Divide both sides by :

Simplify:

### Example Question #12 : Solving By Factoring

Solve for :

Possible Answers:

or

and

Correct answer:

and

Explanation:

For quadratic equations, you need to factor in order to solve for your variable.  You do this after the equation is set equal to zero.  Thus, you get:

Next do your factoring:

into

Then, you set each factor equal to .  Solve each "small" equation:

or  or

BOTH of these are answers to the equation.

### Example Question #13 : Solving By Factoring

Solve for :

Possible Answers:

and

and

Correct answer:

and

Explanation:

For quadratic equations, you need to factor in order to solve for your variable.  You do this after the equation is set equal to zero.  Luckily, this is already done for you!  Thus, start by factoring:

into

Then, you set each factor equal to .  Solve each "small" equation:

or

or

BOTH of these are answers to the equation.

### Example Question #14 : Solving By Factoring

Solve for :

Possible Answers:

or

and

Correct answer:

and

Explanation:

For quadratic equations, you need to factor in order to solve for your variable.  You do this after the equation is set equal to zero.  Thus, you get:

Next do your factoring.  You know that both groups will be positive.  Also, given that the middle term is , you only have one possible choice for your factors of :

Then, you set each factor equal to .  Solve each "small" equation:

or

or

BOTH of these are answers to the equation.

### Example Question #15 : Solving By Factoring

Solve for :

Possible Answers:

and

and

and

Correct answer:

and

Explanation:

For quadratic equations, you need to factor in order to solve for your variable.  You do this after the equation is set equal to zero.  Thus, you get:

Next do your factoring.  You know that both groups will be negative.  This will give you a positive last factor but a negative middle term.  Given the value of the middle term, the factors of  needed will be  and

Then, you set each factor equal to .  Solve each "small" equation:

or

or

BOTH of these are answers to the equation.

### Example Question #16 : Solving By Factoring

Solve for :

Possible Answers:

Correct answer:

Explanation:

This is a quadratic equation, so first, move all terms to the same side by subtracting :

The quadratic polynomial can be factored using the  (or grouping) method. We want to split the middle term by finding two integers with sum  and product ; through some trial and error, we find  and . The equation becomes

Regroup:

Distribute out common factors as follows:

Since the product of these two binomial expressions is equal to 0, one of them is equal to 0; set both to 0 and solve:

Subtract 1 from both sides:

Divide both sides by 3:

or

Add 3 to both sides:

The solution set of the equation is .

### Example Question #17 : Solving By Factoring

If , what could be the value of ?

Possible Answers:

Correct answer:

Explanation:

Start by rearranging the given equation:

Next, factor the equation.

Finally, set each factor equal to zero and solve.

and

Since  can equal to either , we know that  must then equal to either .

### Example Question #18 : Solving By Factoring

Solve for .

Possible Answers:

Correct answer:

Explanation:

Start by factoring the equation.

For this equation, you want two numbers that multiply up to  and add to . The only numbers that fit this criterion are  and .

Thus,

Now, set each of these factors equal to zero and solve.