There’s a lot to learn, and it’s common to devote several weeks to covering key aspects of these functions and their graphs.
Fortunately, only a small number of key concepts and processes must be mastered to ace the SAT/ACT.
Below is a list of the things you need to know about polynomial functions.
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[Note: “iff” means “if and only if.”]
Definition
Polynomial functions: functions of the form p(x) = ax^n+bx^(n–1)+c^(n–2)+…+z.
Polynomial Standard form
y = ax^n+bx^(n–1)+c^(n–2)+…+z.
Factored form
y = a(x–s1)(x–s2)
Let y = 0. Then, s1, s1 … are solutions (a.k.a. “roots”) iff s1, s2, … are x-intercepts (aka “zeros”).
Zeros are values of x for which y equals zero (i.e x-intercepts).
Factors of p(x) iff zeros of p(x) iff roots/solutions of p(x) = 0. That is, “factors,” “zeros,” “roots,” “solutions,” and “zeros” always go together (any one implies all the rest).
Factor theorem
Given polynomial function p(x), c is zero iff x–c is a factor.
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For practice, search Google for worksheets covering any or all of the above, pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this important subtopic.
Fortunately, only a small number of key concepts and processes must be mastered to ace the SAT/ACT.
Below is a list of the things you need to know about polynomial functions.
-
[Note: “iff” means “if and only if.”]
Definition
Polynomial functions: functions of the form p(x) = ax^n+bx^(n–1)+c^(n–2)+…+z.
Polynomial Standard form
y = ax^n+bx^(n–1)+c^(n–2)+…+z.
Factored form
y = a(x–s1)(x–s2)
Let y = 0. Then, s1, s1 … are solutions (a.k.a. “roots”) iff s1, s2, … are x-intercepts (aka “zeros”).
Zeros are values of x for which y equals zero (i.e x-intercepts).
Factors of p(x) iff zeros of p(x) iff roots/solutions of p(x) = 0. That is, “factors,” “zeros,” “roots,” “solutions,” and “zeros” always go together (any one implies all the rest).
Factor theorem
Given polynomial function p(x), c is zero iff x–c is a factor.
-
For practice, search Google for worksheets covering any or all of the above, pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this important subtopic.
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