01Bpolynomials

toc = Polynomials =

A ** polynomial ** is an expression with
 * only one variable (eg x)
 * one or more terms
 * each term has a non-negative, integer power of x

P(x) = a n x n + a n–1 x n–1 + ... + a 2 x 2 + a 1 x + a 0.

In the polynomial, P(x)
 * a n is the coefficient of x n, etc
 * P(x) has a degree of n (the highest power of x)

Polynomials are usually named with consecutive capital letters, starting at P (ie P, Q, R, etc)

** Example 1 **
Q(x) = 2x 3 + x 2 + 4x – 5


 * Q(x) has a degree of 3
 * Q(x) has four terms
 * The variable in Q is x
 * The __constant term__ is –5
 * (this term is ** independent of x ** because it is not affected when x changes)
 * The value of Q(1) is the result of substituting x = 1 into Q
 * Q(1) = 2(1 3 ) + 1 2 + 4(1) – 5
 * Q(1) = 2

** Example 1 on the Classpad **
We can define and use polynomials on your calculator

In the MAIN screen, go to the ACTION menu, COMMAND submenu and select **Define**. {The word **Define** should appear on your screen} On the same line as **Define**, type q(x) = 2x^3 + x^2 + 4x – 5 {Use the **ABC** tab from the virtual keyboard to enter q, __but__ use the variable button and **NOT** the ABC tab to enter the variable, //**x**//}

Press **EXE** and the polynomial q(x) is now in your calculator.

Now enter q(1) then **EXE** and you should get the value 2

We can also manipulate q in other ways, for example, try entering 3q(x)

And then try entering
 * expand**(3q(x))

{The **expand** function is in the ACTION menu, TRANSFORMATION submenu}

Adding Polynomials
We add (or subtract) polynomials simply by adding (or subtracting) the like terms.

**Example 2**
If P(x) = x 3 + 2x 2 – 5x – 3 and Q(x) = x 4 + x 3 – 2x 2 – x + 5

Then P + Q = x 4 + 2x 3 – 6x + 2

P – Q = – x 4 + 4x 2 – 4x – 8

**Example 3**
Given the polynomial: P(x) = x 3 + ax 2 + bx – 4 where a and b are constants and given that P(1) = – 2 and P(2) = 2

Find the values of a and b

P(1) = –2 1 3 + a(1 2 ) + b(1) – 4 = –2 1 + a + b – 4 = –2 a + b = 1 ** → (1) **

P(2) = 2 2 3 + a(2 2 ) + b(2) – 4 = 2 8 + 4a + 2b – 4 = 2 4a + 2b = –2 ** → (2) **

Now solve simultaneously 2x**(1)** 2a + 2b = 2 ** → (3) **

**(2)** – **(3)** 2a = –4 a = –2

sub a = –2 into ** (1) ** –2 + b = 1 b = 3


 * Thus a = –2 and b = 3 **

** Example 3 on the Classpad **
Your calculator can solve simultaneous equations like this pair.

Go to the ACTION menu, COMMAND submenu and select **Define** On the same line as **Define**, Type p(x) = x^3 + ax^2 + bx – 4 {use the **ABC** tab of the virtual keyboard for p, a, b and use the **variable x** button}

Press **EXE** and your screen should look like figure 2.

Now go to the **2D** tab and select the icon circled in figure 3. An empty form should appear on your screen in that shape.

Enter the following into the three empty boxes on the form as you see in figure 3: p(1) = –2 p(2) = 2 a, b {Use the ABC tab for p, a, b}

Press EXE and you should get the correct answer {a = –2, b = 3}

Go to top of page flat

.