Factoring

A polynomial like x² + 6x + 14 that cannot be factored is like a prime number in that its factors are only what? We say that such an expression is ___________.

1 and itself/prime

What are some ways to factor an expression?

1. By using factor trees which will show only the prime factors. 2. By listing which will show all factors.

What is the prime factorization method for finding the GCF?

1. Write the prime factorization of each given number in exponential form. 2. Create a factorization for the GCF that includes only those prime factors common to all the factorizations, each raised to its smallest exponent in the factorization. 3. Multiply

How can we check to see if our factored form of an expression is correct?

By multiplying it using the distributive property. If the result is the original polynomial, then our factorization is correct.

What is the major drawback about using trial and error to factor? What is an alternative way to factor that is not so tedious?

It can be tedious to find the correct binomial combination. An alternative way to factor is to factor by grouping.

What is factoring by grouping?

It is a technique that is used to factor a four-term polynomial such as xy + 2y + 7x + 14. The method is called grouping because we group pairs of terms together and then look for a common factor within each group. 1. Factor out any monomial GCF (other than 1) that is common to all four terms. 2. Group together pairs of terms and factor the GCF out of each pair. 3. If there is a common binomial factor, then factor it out. 4. If there is no common binomial factor, then interchange the middle two terms and repeat the process. If there is still no common binomial factor, then the polynomial cannot be factored by grouping.

What is factoring?

It is another way to rewrite a numerical or algebraic expression by writing it as a product of prime factors. It is the reverse of multiplying. The process is used extensively in simplifying the more complex expressions out there.

How do you factor trinomials of the form ax² + bx + c by grouping?

Since a trinomial of this form has only three terms, we will split the bx term into two like terms to create a four-term polynomial that we can then factor by grouping like normal when four polynomials exist. To determine how to split up the bx term, we use the fact that if ax² + bx + c is factorable, then b will always equal the sum of a pair of factors of the product of a and c.

How do you factor using a factor tree?

The idea is to draw two branches beneath any composite number and write two factors of that number at the end of the branches. You keep going until you no longer have any composite numbers. Note: Many people circle each prime factor to make them stand out.

What is the greatest common factor?

The largest natural number that divides all given numbers evenly.

How do you factor trinomials of the form ax² + bx +c?

These forms also have two binomial factors. When a is not 1, we must consider the factors of the ax² term and the factors of the c term. We have to develop a trial and error method for finding the factored form:

How do you factor by listing all factors?

We can divide whatever number you are trying to factor by 1,2,3 and so on, writing each divisor and quotient pair as a product until we have all possible combinations. When you reach a natural number that you already have listed, you can stop.

How do you factor trinomials of the form x²+bx+c?

When we factor a trinomial of this form, we are reversing the FOIL process. To factor x² + bx + c: 1. Factor out a monomial if needed. 2. Find two numbers with a product equal to c and sum equal to b. 3. The factored trinomial will have the form: (x + first number)(x + second number). Note: The signs in the binomial factors can be minus signs, depending on the sings of b and c. If the third term is negative, then the two binomials will have opposite signs. The sign of the number with the greatest absolute value of the two binomials will be the same as the sign of the middle term of the trinomial. If the middle term is negative and the third term is positive in the trinomial, then both of the binomials of its factored form will be negative. If both the middle term and third term in the trinomial are positive, both binomials in its factored form will be positive.

Sometimes, when factoring, the GCF is a polynomial. How do you factor these?

You factor them out just like any other GCF

When factoring polynomials, the first step is to determine what?

if there is a monomial factor that is common to all the terms in the polynomial. Further, we will want that monomial factor to be the greatest common factor of all of the terms.

When factoring a polynomial where the first term is negative, it considered undesirable to have the first inside the parantheses ______________. How can we avoid this?

negative/by factoring the negative of the GCF; where the GCF will have a negative sign instead of the first term inside the parentheses.

The listing method is good method for ____________ numbers where as _____________ factorization is better for larger numbers.

smaller/prime

No matter what type of polynomial we are asked to factor, we will always consider what?

whether a monomial GCF (other than 1) can be factored out of the polynomial.