Unit 2 - Functions
Required Prior Knowledge
Questions
Expand and Simplify
a) \(\left(x+3\right)\left(2x-1\right) \)
b) \(\left(x-2\right)\left(x^{2}+2x+4\right)\)
Using long division, compute \(2149\div 19\) giving your answer with a remainder.
Do you know multiple calculation methods for this division?
Solutions
Get Ready
Questions
What does the number \(537\) represent?
Solutions
Notes
A polynomial is a function of the form$$P\left(x\right)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_{r}x^{r}+\cdots+a_{1}x+a_{0}$$with \(n,r\in\mathbb{N}\), where:
\(x\) is the variable
\(a_{0}\) is the constant term
\(a_{n}\) is the leading coefficient
\(a_{r}\) is the coefficient of \(x^{r}\)
\(n\) is the order or degree of the polynomial
We add and subtract polynomials by collecting like terms.
To multiply a polynomial by a scalar (or constant) we distribute over the bracket. That is, we ‘expand the bracket’.
To multiply a polynomial \(P\left(x\right)\) by another polynomial \(Q\left(x\right)\) we must multiply ALL terms in the first polynomial by ALL terms in the second polynomial.
There are multiple methods to do this, but you must ensure you do not miss any possible pairs. The grid method is a good way to approach this.
Examples and Your Turns
Example
Let$$P\left(x\right)=x^{3}-2x^{2}+3x-5$$and$$Q\left(x\right)=2x^{3}+x^{2}-11$$Find
(a) \(P\left(x\right)+Q\left(x\right)\)
(b) \(P\left(x\right)-Q\left(x\right)\)
(c) \(2P\left(x\right)\)
(d) \(-3Q\left(x\right)\)
(e) \(P\left(x\right)\times Q\left(x\right)\)
Your Turn
Let$$P\left(x\right)=x^{3}-2x+4$$and$$Q\left(x\right)=2x^{3}+3x-5$$Find
(a) \(P\left(x\right)+Q\left(x\right)\)
(b) \(P\left(x\right)-Q\left(x\right)\)
(c) \(3P\left(x\right)\)
(d) \(-Q\left(x\right)\)
(e) \(P\left(x\right)\times Q\left(x\right)\)
Notes
A zero of a polynomial is a value of the variable which makes the polynomial equal to zero.
i.e. \(\alpha\) is a zero of \(P\left(x\right)\) if and only if \(P\left(\alpha\right)=0\)
The roots of a polynomial equation are the solutions of the equation \(P\left(x\right)=0\).
i.e. \(\alpha\) is a root of \(P\left(x\right)=0\) if and only if \(P\left(\alpha\right)=0\)
The \(x\)-intercepts of the graph of \(y=P\left(x\right)\) are the points where the graph crosses the \(x\)-axis.
These essentially tell use the same thing, but the wording varies depending on context.
For example, consider \(P\left(x\right)=x^{3}-2x^{2}+x-12\).
Then \(P\left(3\right)=\left(3\right)^{3}-2\left(3\right)^{2}+\left(3\right)-12=0\).
This tells us that:
\(3\) is a zero of the polynomial \(x^{3}-2x^{2}+x-12\);
\(3\) is a root of the equation \(x^{3}-2x^{2}+x-12=0\);
the graph of \(y=x^{3}-2x^{2}+x-12\) has an \(x\) intercept at \(\left(3,0\right)\).
Two polynomials are equal if and only if:
they have the same degree/order
all corresponding terms have the same coefficients
If we know that two polynomials are equal, then we can equate coefficients to find missing coefficients.
Examples and Your Turns
Example
If \(\left(ax\right)^{2}+bx+3\left(ax^{2}+x\right)=4x-2x^{2}\), find the values of \(a\) and \(b\).
Your Turn
Given that \(\left(x^{2}+ax-3\right)\left(x^{2}-x+b\right)=x^{4}-cx^{2}+9\) find the values of \(a,b\) and \(c\).
Your Turn
Given that \(4x^{3}+10x^{2}-x+15=\left(x+3\right)\left(ax^{2}+bx+c\right)\) find the values of \(a,b\) and \(c\).
Your Turn
\(\left(x-1\right)\) is a factor of \(x^{3}+3x^{2}+2x-6\). Find all remaining factors.
Your Turn
\(\left(x+3\right)\) is a factor of the polynomial \(x^{3}+ax^{2}-7x+6\). Find the value of \(a\), and hence the remaining factors.
Your Turn
\(\left(2x+3\right)\) and \(\left(x-1\right)\) are factors of \(2x^{4}+ax^{3}-3x^{2}+bx+3\).
Find the constants \(a\) and \(b\), and all zeros of the polynomial.
Notes
Calculate the division \(2798\div 12\) giving your answer as a number with a remainder.
This means we can write $$2798=233\times 12 + 2$$or equivalently$$\frac{2798}{12}=233+\frac{2}{12}$$
The following terminology is used for the four numbers in this division:
The dividend is \(2798\)
The divisor is \(12\)
The quotient is \(233\)
The remainder is \(2\)
The same terminology is used when dividing polynomials.
Consider $$2x^{3}+7x^{2}+9x+8=\left(2x^{2}+3x+3\right)\left(x+2\right)+2$$This means that$$\frac{2x^{3}+7x^{2}+9x+8}{x+2}=2x^{2}+3x+3+\frac{2}{x+2}$$
Can you notice the similarities between this and the result from the calculation of \(2798\div 12\)?
If a polynomial \(P\left(x\right)\) is divided by another polynomial of lesser degree, \(D\left(x\right)\), then we can write$$P\left(x\right)=Q\left(x\right)\times D\left(x\right) + R\left(x\right)$$where \(Q\left(x\right)\) is the quotient and \(R\left(x\right)\) is the remainder.
We can also write this as$$\frac{P\left(x\right)}{D\left(x\right)}=Q\left(x\right)+\frac{R\left(x\right)}{D\left(x\right)}$$
Examples and Your Turns
Example
Divide \(x^{3}-11x+3\) by \(x+5\). Hence write \(x^{3}-11x+3\) in the form \(Q\left(x\right)\times\left(x+5\right)+R\left(x\right)\).
Your Turn
Show that$$\frac{2x^{3}+7x^{2}+10x+15}{x+2}\equiv 2x^{2}+3x+4+\frac{7}{x+2}$$by performing the division.
Your Turn
Divide \(2x^{3}+x^{2}+5x-1\) by \(2x-1\).
Hence write \(2x^{3}+x^{2}+5x-1\) in the form \(Q\left(x\right)\times \left(2x-1\right)+R\left(x\right)\).
Your Turn
Divide \(x^{4}+2x^{2}-1\) by \(x+3\).
Hence write \(x^{4}+2x^{2}-1\) in the form \(Q\left(x\right)\times \left(x+3\right)+R\left(x\right)\).
Example
Divide \(x^{4}+4x^{3}-x+1\) by \(x^{2}-x+1\).
Hence write \(x^{4}+4x^{3}-x+1\) in the form \(Q\left(x\right)\times \left(x^{2}-x+1\right)+R\left(x\right)\).
Your Turn
Divide \(3x^{4}+4x^{3}+2x^{2}+3x-1\) by \(x^{2}+x-2\).
Hence write \(3x^{4}+4x^{3}+2x^{2}+3x-1\) in the form \(Q\left(x\right)\times \left(x^{2}+x-2\right)+R\left(x\right)\).
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
Why are \(\sqrt{x}\) and \(x^{-2}\) not considered polynomial terms?
In polynomial division, why must the degree of the remainder be less than the degree of the divisor?
Explain the power of equating coefficients.
Common Mistakes / Misconceptions
Errors in the process of long division are the most common (though we will see ways to avoid this in the next skill).
Assuming the remainder is always \(0\).
Connecting This to Other Skills
Working with algebraic fractions (PK5) is an important prior knowledge skill.
These ideas will build into the remainder and factor theorems (2.20) where we will see other methods to compute remainders and find factors.
This is vital in finding the roots of polynomials, which we see more of in the conjugate root theorem and fundamental theorem of algebra (2.21), and also the sum and product of roots (2.22).
Working with polynomials is a core skill for calculus, both differentiation (unit 4) and integration (unit 5).
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?