# Synthetic division,Remainder factor theorem, graphing cubic

1. Explain what synthetic division is and what it is used for. (include at least 2 different uses for synthetic division) Give an example of synthetic division, show all steps. Explain what your answer means.

2. Pick a cubic function. (use something not too simple. a good example: y = 2(x+1)^3 - 5) Starting with y = x^3, use vertical and horizontal translations, as well as, shrinking and stretching to graph your function. (make sure to explain in English what you are doing as well as graphing y = x^3 and your function) Using an algebraic method, find the roots of your function. Describe the symmetry of your function. Explain the concept of symmetry in general. (which functions are symmetric, which are not. how can you tell from the equation?)

3. State and explain (in simple English) what Remainder and Factor Theorems mean.

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1. Explain what synthetic division is and what it is used for. (include at least 2 different uses for synthetic division) Give an example of synthetic division, show all steps. Explain what your answer means.

Synthetic division is a shortcut method for dividing a polynomial by a linear factor of the form x + a or x - a.

Synthetic division can also be used in conjunction with the "Remainder Theorem" to find the value of a polynomial at a real value.

Example:

Let us take the polynomial 3x^2 + 4x + 4 divided by x - 1

Next, all the variables and their exponents ( , ) are removed, leaving only a list of the coefficients: 3, 4, 4. These numbers form the dividend. We form the divisor for the synthetic division using only the constant term (1) of the linear factor as shown below:

The first number in the dividend 3 is put into the first position of the result area (below the horizontal line). This number is the coefficient of the x^2 term in the original polynomial:

Then this first entry in the result (3) is multiplied by the divisor (1) and the product is placed under the next term in the dividend (4):

Next the number from the dividend and the result of the multiplication are added together and the sum is put in the next position on the result line:

This process is continued for the remainder of the numbers in the dividend:

This means that

P(x) =

OR value of the polynomial P(x) at x = 1 is 11.

2. Pick a cubic function. (use something not too simple. a good example: y = 2(x+1)^3 - 5) Starting with y = x^3, use vertical and horizontal translations, as well as, shrinking and stretching to graph your function. (make sure to explain in English what you are doing as well as graphing y = x^3 and your function) Using an algebraic method, find the roots of your function. Describe the symmetry of your function. Explain the concept of symmetry in general. (which functions are symmetric, which are not. how can you tell from the equation?)

Solution:

Let us take the cubic function

First graph y = x^3 the graph is shown below:

Then shift this graph 2 units to the right to graph y = (x - 2)^3. Graph is shown below:

Then slide this graph 8 units down to graph . Graph is shown below:

To find the factors of put y = 0, we will get

Which gives and

Roots are , 4

It is clear from the graph of the function that it is not symmetric about x-axis, y-axis and origin.

HOW TO TEST FOR SYMMETRY:

x-axis symmetry: A graph of an equation is symmetric with respect to the x-axis if a substitution of -y for y leads to an equivalent equation.

y-axis symmetry: A graph of an equation is symmetric with respect to the y-axis if a substitution of -x for x leads to an equivalent equation.

Origin symmetry: A graph of an equation is symmetric respect to the origin of the simultaneous substitution of -x for x and -y for y leads to an equivalent equation.

3. State and explain (in simple english) what Remainder and Factor Theorems mean.

Remainder Theorem:

The remainder theorem states that if a polynomial P(x) is divided by x - a , the remainder is P(a).

Suppose we take polynomial p(x) = x^2 + 4 and divide it by x - 4 then remainder will be p(4).

Factor Theorem:

For any polynomial P(x), x - a is a factor of P(x) if and only if P(a) = 0.

That means If we divide a polynomial p(x) by a factor x - a of p(x), then you will get a zero remainder.

Instead if we take r(x) = 0 then p(x) = (x - a)q(x) which shows (x -a) is a factor of p(x).

Example:

(x -2) is a factor of polynomial p(x) = (x^2 - 4) iff p(2) =0

Here p(2) = 2^2 - 4 = 4-4 = 0

So (x -2) is a factor of polynomial p(x) = (x^2 - 4).

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