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Vocabulary of Quadratic Polynomials - Concept
Adding and subtracting rational expressions is similar to adding fractions. When adding and subtracting rational expressions, we find a common denominator and then add the numerators. To find a common denominator, factor each first. This strategy is especially important when the denominators are trinomials.
Math problems can be tricky if you don't understand the directions. And that's especially tricky when you come to your study of quadratic functions because I'm going to show you four different instructions that essentially mean the same thing. Check it out.
Find the roots of the quadratic polynomial f of x equals x squared plus 3x plus 2. If you saw those directions what you would do is set this function equal to zero, factor or use the quadratic formula or complete the square or do graphing choose whatever method you want to to solve this. I'm going to factor because I'm a pretty good factorer. And I get the solutions x=-1 and x=-2 when I use the zero product property. So that's one set of instructions for this problem.
Let's look at the same problem with a different set of instructions. Find solutions of the equation. Notice the difference. This was roots of a polynomial. This is solutions of an equation. It's the same mathematical process. I'm still going to go ahead and factor, use the zero product property and I'm going to get the same answers. x=-1, x=-2, go back check your solutions by substituting them in if you want to. That's tricky. Same problem, different words, let's look there's even a third way to do it. Let me back up.
What we have here is find the x intercepts of the graph of y=x squared plus 3x plus two. Notice this is the same equation, the same function, the same polynomial only now they're telling me to look for the x intercepts. So if I look on the x intercepts, here is where I get my answers -2 and -1. It's the same answers. What I really I'm trying to show you guys that these four things are all just different instructions for doing the same mathematical process.
Roots of polynomials, solutions of equations, intercepts of graphs and we also have zeros of a function. And what would happen with zeros of a function is you would have a function like this, this is the same one we've been working with all along. I'm looking at the table and finding the zeros. Finding the zeros means look for where your y value or in our case we're looking at f of x is equal to zero. Here they are right here. -2 and 1, does that sound familiar? It's the same answer we got all along. So you guys, I understand. Like I'm a Math teacher, I get it. I do all these problems with you. I know the directions can be really tricky and that's why I wanted to try to clarify. These are four different sets of instructions using four different sets of vocabulary pairings that essentially mean the same thing.
One thing I want you guys to start thinking about is how they're alike and how they're different, and also thinking about what vocabulary words go together. Graphs goes with x intercepts, roots goes with polynomials, solutions goes with equations. Those kinds of pairings, if you mess them up, people would still understand what you're talking about but formally you want to make sure you have the right match ups when you're doing and talking about your Math problems.