You are here:

Basic Math/factors and integer coefficients

Advertisement

Expert: Josh - 12/15/2009


Question

the problem
the math problem is attached. i am not sure if i put in numbers for x and y, such as what is y when x is zero and vise vera, but it's not asking that so... this is where you come in!
thanks so much and happy holidays!

Answer
The question reads:

Put the function into factored form with integer coefficients and then identify any horizontal intercepts.

y = -10t^2 +225t

[Write your points in ascending order of its x-coordinates. If there is only one horizontal intercept, put the same coordinates for both points]

(   ,   ) and (   ,   )

--------
Hi Maddie,

The concept of factorization can be taught in a number of ways. I can take a guess, but cannot be 100% sure how you would fill-in the blanks [I am referring to the parentheses with space separated by a comma] at your school.

To make things interesting, let us consider y=10*t^2+225*t instead.
The standard way to do it is as follows:

Observations:
1) "t" appears in both 10*(t^2) and 225t, so it is a common factor.
2) Both 10 and 225 are divisible by 5 [in fact 5 is the greatest common factor].

So, it would be appropriate to pull out a factor of "5t" from the expression.

| Brief Review:
| [Quotient] Recall that when we perform a division like 15/3,
|   the answer we are seeking (i.e., 5 here) is called the "quotient";
|   15 and 3 are referred to as the dividend and divisor, respectively.
| [Division involving integers and an unknown variable]
|   Consider 6*(t^2) / 2t. We just treat those parts containing numerals
|   and variables separately. For the numerals, 6/2=3. For the variable
|   (all things containing "t"), (t^2)/t=t. So, altogether,
|   6*(t^2) / 2t = 3t.

In your head, you divide 10*(t^2) by 5t to get 2t. Similarly, you divide 225t by 5t to get a quotient of 45. Now, the original expression may be factorized as

10*(t^2) + 225t
= 5t (2t + 45)

Note: Here, our unknown variable is "t", not "x".

| Brief Review:
| [Meaning of vertical intercept]
| 1. In a straight line equation, y=m*t+b, the vertical axis
|    represents "y" while the horizontal axis represents "t".
|    In this instance, "b" is called the y-intercept (a.k.a.
|    vertical intercept) precisely because this is the point
|    where it crosses the y-axis. This happens when we put t=0
|    in the equation.
| [Meaning of horizontal intercept]
| 2. Consider a function of "t", f(t) [which we can also call "y"].
|    If y can be factorized as (t-a)(t-b) for some number
|    "a" and "b", then y=(t-a)(t-b) equals zero, whenever
|    t=a, or t=b. Compare to the previous paragraph, here, we are
|    setting y=0, instead of setting t=0. Graphically, when y=0,
|    we are basically considering the point(s) where a curve
|    crosses the t-axis. Since the t-axis here represents the
|    horizontal axis, these points t=a and t=b are referred to
|    as horizontal intercepts.

The question asks us to order the factors in ascending order (from smallest to largest) with respect to their t-intercepts. First, we can think of 5t as (5t-0). Clearly, this can only be zero when t=0. i.e., t=0 is our first horizontal intercept. Second, (2t+45) plays a similar role to (t-b) in our previous discussion (see point 2). Again, we want to find out when (2t+45) equals 0. The solution is t=-45/2 or t=-22.5. Note carefully that our second horizontal intercept, viz., t=-22.5, has larger absolute magnitude then our first horizontal intercept, viz., t=0. Nonetheless, t=-22.5 appears to the left of the origin (t=0) on a number line. So, the proper order for the factors is (2t+45)(5t-0).

To do exactly what the question asks, we want an integer coefficient up the front. So, we have
10*(t^2) + 225t
= 5t (2t+45)
= 5 (2t+45)(t-0)

Although you can probably get away writing this in the form of "5(2,45)(1,0)", this is not recommended as this is not a standard convention. Certainly, this notation is not universally accepted nor widely understood. You can probably write it this way, if this is how you have been taught at school, provided that its definition has been properly introduced. But do not expect that anyone outside your school community (or class) will necessarily be able to follow what this actually means. Think of it as more of a short-hand which you may consider using when things get repetitive and no one other than your teacher is going to read it.

Note: I haven't actually answered the question you originally asked. The difference lies in having a minus sign (-10 instead of 10), so the ordering of the factors will be different. This is left for you to do.

Enjoy the festive season!

Josh

Expertise

When I work through problems, I like to emphasize concepts which I believe are worth noting. I will try to answer questions in the following areas, but not at the advanced level. Algebra. Sequences & Series. Trigonometry. Functions & Graphs. Coordinate Geometry. Quadratic Polynomials. Exponential & Logarithms. Basic Calculus. Probability, Permutation and Combination. Mathematical Induction. Complex numbers. Physics problems.

Experience

I have worked as a teaching assistant in college. My hope is that more people will share knowledge without boundary, give help without seeking recognition or monetary rewards.

Education/Credentials
Bachelor degree in Engineering Science

©2012 About.com, a part of The New York Times Company. All rights reserved.