Factoring Trinomials of the Form x^2 + bx + c

In this section we will factor trinomials - polynomials with three terms. Students find this difficult at first. However, with much practice factoring trinomials becomes routine. If a trinomial factors, then it will factor into the product of two binomials.

Factor:

   Step 1: Factor the first term: x^2 = x*x.

   Step 2: Factor the last term. Choose factors that add or subtract to obtain the middle term.

   Step 3: Determine the signs by adding or subtracting the product of the inner and outer terms.

   Step 4: Check by multiplying.

Rather than trying all possible combinations of the factors that make up the last term spend some time looking at the factors before starting step two. Look for combinations that will produce the middle term. Here is the thought process in choosing 3 and 4 in step two above:

   "Can I add or subtract 1 and 12 to obtain 7?" – NO     

   "Can I add or subtract 2 and 6 to obtain 7?" – NO    
   "Can I add or subtract 3 and 4 to obtain 7?" – YES, because +3 + 4 = +7

Factor the trinomials.

This process used for factoring trinomials is sometimes called guess and check or trial and error. The biggest problem occurs when the signs are improperly chosen. With this in mind, you should take care to check your results by multiplying. Also, since multiplication is commutative order does not matter, in other words

If the trinomial has a GCF you should factor that out first.  Also, you should factor in such a way as to ensure a resulting trinomial with a positive leading coefficient.

Factor the trinomials.

Take care to perform the check. Most of the problems that you will encounter factor nicely but be sure to watch out for something like this . The middle term works but the last term does not:

because the sign of the last term is incorrect

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Introduction to Polynomials

Before beginning this section we first make a few definitions. Some of the terms have already been defined but it does not hurt to go over them again. First recall that algebraic expressions such as,

are called terms. The term,

has a coefficient −2 and a variable part a^2 b. A polynomial is any sum or difference of algebraic terms. A polynomial with only one term is called a monomial. A polynomial with 2 terms is called a binomial and one with 3 terms is called a trinomial.  We will not use any other special names for polynomials with more terms.

It is common practice to express polynomials in descending order from largest exponent down to the constant term. The constant term, or the term with no variable part, can be thought of as the coefficient of the x^0 term.

The degree of a polynomial with one variable is the largest exponent.

Evaluating expressions involves replacing the variable with the appropriate numerical value. In other words, plug in the values and use the order of operations to calculate the answer.

Evaluate.

Simplifying expressions first saves steps when evaluating and the results will be the same.

We can use function notation to evaluate. Do not let the notation get in the way of your ability to do these types of problems. The idea is the same, just substitute in the appropriate values.

Evaluate.

We will often run into problems where the graph is given instead of the algebraic equation. In this case, read the graph and answer the question.

Given the graph of the function.

Given the graph of the function.

Projectile Problem: A projectile is fired from the ground with an initial velocity of 64 feet per second. The height of the projectile in feet after t seconds is given by the function

  with the following graph:

   a. Use the graph to determine how much time it takes to reach the maximum height.
   b. How much time will it take to hit the ground?
   c. What times will the projectile be at 60 feet?
   d. Use the function  to determine the height of the projectile at t = 1 second.

Read the graph to answer a, b, and c.

Geometry Problem: The function

gives the volume of a sphere given its radius r. Use the function to calculate the volume of a sphere of radius 6 centimeters. (Use 3.14 as an approximation for pi.)

Projectile Problem: A projectile is fired upwards from the roof of a 256 foot building with an initial velocity of 96 feet per second. The height of the projectile in feet after t seconds is given by

whose graph is given:

   a. How much time does it take to reach a maximum height?
   b. What is the maximum height?
   c. How much time does it take for the projectile to hit the ground?
   d. Use the function to determine the height after 6 seconds.

Read the graph to answer a, b, and c.

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