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.

Solution Preview

Please find the solution/explanation attached herewith.

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 ...

Solution Summary

A step-by-step explanation are provided for synthetic division, graphing cubic functions using transformations, and remainder factor theorem.

(i) Determine the remainder when 9x^5 â?" 4x^4 is divided by 3x â?" 1.
(ii) Show, using the factortheorem, that 2x â?" 1 is a factor of
2x^4 â?" x^3 â?" 6x^2 + 5x â?" 1
and hence express 2x^4 â?" x^3 â?" 6x^2 + 5x â?" 1 as a product of a linear
and cubicfactor.
SEE ATTACHED FILE:

How would you explain the performance of synthetic division to someone else? Use the problem (2x3 - 3x2 - 11x + 7)/(x + 3) to support your explanation.

Note the cubic equation x3 - 6x2 + 11x - 6 = 0. I will claim that x = 1 constitutes a root of that equation ( replace x by 1 in the equation to verify ). Thus, the binomial (x - 1) will divide our cubic equation evenly, without remainder. Perform the division to obtain the quotient, which is a quadratic equation. Completely

1) Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and express P in the form P(x) = D(x) · Q(x) + R(x).
P(x) = 3x^2 + 4x â?' 1, D(x) = x + 5
P(x) =(x+5)(_____)+______
2) Two polynomials P and D are given. Use either synthetic or long division to divide P(x) by D(x), and

Please help me with the attached problems by explaining how to solve them.
Use synthetic division to divide the polynomial 4x3 + 10x2 - 11 by x + 3, and state the quotient polynomial and the remainder. [Be careful - notice that there is no x term.]. Show work.
Consider the polynomial f(x) = 5x3 - 9x2 - 17x - 3.
(a) By u

1) Determine the value of k so that when P(x) = x(cubed) + kx(squared) - 2x(squared) + 1. Is divided by x + 2, the remainder is 5
2) Using long division, determine the remainder when P(x) is divided by x-3

1. Explain how synthetic division may be used to find the factors/zeros of a polynomial function. Give an example of how this is accomplished.
Use synthetic division to find the function value.
1) f(x) = 2x4 + 4x3 + 2x2 + 3x + 8; find f(-2).
Write the quadratic function in the form y = a(x - h)2 + k.
2) y = x2 - 2x - 9

1. FIND ALL VERTICAL ASYMPTOTES OF THE FUNCTION.
*******X+5
F (x) -------------------
****4x squared+7
2. WHICH SHOWS THE TRUE STATEMENT FOR THE GRAPH OF THE RATIONAL FUNCTION g.
*******X+2
g(x) -------------------
****x squared+2x-3
3.USE SYNTHETIC DIVISION TO FIND UPPER AND LOWER BOUNDS OF THE REAL ZEROS OF f.