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Polynomial Functions

6.1 Antiderivatives of Functions

Antiderivatives of Specific Functions

• We will be looking at antiderivatives of polynomial, exponential and trigonometric functions.
• The formulas are as below:
i) Polynomial functions

\int x^{n} d x=\frac{1}{n+1} x^{n+1}+c \quad,\ n \neq-1

\int x^{-1} d x=\int \frac{1}{x} d x=\ln |x|+c

Note: Notice that in there is an absolute value over to make the result fit the domain for all logarithmic functions is (0,\ +\infty), and x=0 is not possible in the first place as it would make \frac{1}{x} undefined.

ii) Exponential functions
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3.1 Differentiation Rules for Polynomials

Differentiation Rules for Polynomials

• The other rules regarding differentiation such as addition, subtraction, multiplication, division of functions, chain rule etc. applies too.
• All of these rules can be proven via first principles, but it is not required.
• The only rule for differentiating polynomials:

\text { If } f(x)=x^{n}, \text { then } f^{\prime}(x)=n x^{n-1}

where n \neq 0, n \in R (or n \in R \backslash\{0\})

Note: If n=0, then f(x)=a \text { constant, } f^{\prime}(x)=0

Proof (Not required)

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6.1 Revision for Linear Systems

Linear Literal Equation Systems

• In previous years, you are already equipped with the skills to solve simple linear equations like 4 y-\frac{(3 y+4)}{2}+\frac{1}{3}=\frac{5(4-y)}{3}, including their corresponding worded problems. The skills learnt are crucial and will be applied in the topics explored in this course.
• A literal equation in x is an equation whose solution will be expressed in terms of pronumerals rather than numbers.

In simpler terms, instead of solving

2x+5=7

instead you will be dealing with

ax+b=c

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5.2 Simultaneous Equations: Polynomials

Solving Equations

• Every now and then, you will be challenged with questions to solve .
• Suppose we have y=f(x) and y=g(x) as two separate functions. If f(x)=g(x), it means that the graph of both functions intersect. Note that this tells nothing about the number of possible values for x.

Simultaneous Equations: Polynomials

• We will first look at how this type of questions looks like, with polynomials first.
• The steps are rather simple:
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2.2 Remainder Theorem, Factor Theorem and the Rational-Root Theorem

Remainder Theorem and Factor Theorem

Now we are going to set D(x)=x-a. Therefore, as \operatorname{deg}(R)<\operatorname{deg}(D), R(x) must be a constant, which we denote as R.

Rewriting the equation gives

P(x)=(x-a) Q(x)+R

• Simply set x=a gives P(x)=(a-a) Q(x)+R=0+R=R. Thus, R=P(a).
• Now we generalize D(x) into any linear polynomial, i.e. D(x)=a x+b, where a,\ b are integers. Therefore, now P(x)=(a x+b) Q(x)+R and substitute x=-\frac{b}{a} (to make D(x)=0) gives
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2.1 Polynomial Function Introduction and Algebraic Operations

Polynomial Functions Basics

• A polynomial function is a function that can be written in the form

y=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}

where n \in N \cup\{0\} (as in non-negative whole numbers) and a_{0},\ a_{1},\ \ldots,\ a_{n} \in R with a_{n} \neq 0.

• The number 0 is called the zero polynomial.
• The leading term, a_nx^n, of a polynomial is the term of highest index among those terms with a non-zero coefficient.
• The degree of a polynomial is the index of the leading term, as in, n.
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5.1 Applications of Polynomial Functions

Applications of Polynomial Functions

• Polynomial functions are a very common and useful function that are used in daily life. Below are a few examples.
• Note that the function might be piecewise. As in, it might be a polynomial for a certain range of the domain, and another polynomial in the other (or it might be other functions like exponential or trigonometric instead).
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3.5 Transformations of Polynomial Functions

Transformations with Power Functions

• Recall that power functions has the form y=x^n, where is a real number. We have looked into cases where is a rational number, which includes positive and negative whole numbers and fractions.
• Transformation can be applied on power functions. The transformed equation simply take the general form of f(x)=a(b(x-c))^{n}+d, where a,\ b,\ c,\ d are real numbers.

i) If it is a reflection, we have b=-1 or a=-1.

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2.2 Cubic Function and Higher Power Polynomials

Cubic Functions Basics

• Cubic functions has the general form

f(x)=a x^{3}+b x^{2}+c x+d, where a,\ b,\ c,\ d \in R,\ a \neq 0

• Cubic functions cannot be fully understood without the use of calculus. Therefore, it will be revisited in A3 – Calculus.
• The ‘shapes’ of cubic graphs vary. Below is a gallery of cubic graphs, demonstrating the variety of ‘shapes’ that are possible.
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