Synthetic Division

We learned in class that Synthetic Division is a shorter method of polynomial division. When using this method, the arrangement of the coefficients of f(x) should be in a descending order of power and substitute the missing terms with zero. 

    Example: 
          
      Divide 3x³ – 2x² + 3x – 4 by x – 3 
         

•First, write down all the coefficients, and put the zero from x – 3 = 0 (so it`ll be x = 3) at the left.
•Next, 'drop' the first coefficient.
•Multiply the number by the potential zero (which is 3), carry up to the next column, and add down.
•Repeat this process.
•Repeat this process again until you reach the end.

Hence, we`ll get 68 as our remainder. Because we began with a polynomial of degree 3 and then divided by x – 3, we are left with a polynomial of degree 2. Then the bottom line symbolized the polynomial 3x<sup>2</sup> + 7x + 24 with a remainder of 68. Put the final result into the required "mixed number" format, we`ll get the answer:

Another example:

         Divide 

                  

Long Division

Before you divide anything, you must identify the value or expression that corresponds to:

• The divisor
• The dividend
• The quotient
• The remainder

Steps on dividing polynomials:

1. Write the dividend and divisor polynomial in descending powers of the literal variable
2. Divide the leading term of the dividend by the first term of the divisor to obtain the first term of the quotient
3. Multiply the divisor by this newly formed term term of the quotient using the Distributive Law and subtract the result from the dividend
4. Treat this remainder, obtained in step 3, along with the rest of the dividend as the new dividend and repeat steps 2 and 3 until the remainder is a degree lower than the divisor. 

In conclusion, we write down the result of the polynomial in x, P(x), by a binomial of the form x - a, is P(x) / x - a = Q(x) + R / x - a, where Q(x) is the quotient and the R is the remainder. You could also write it as  P(x) = Q(x) * (x - a) + R.

x^3 + 7x^2 + 7x - 6 / x + 2 = x^2 + 5x - 3

or

x^3 + 7x^2 + 7x - 6 = (x^2 + 5x - 3)(x + 2)

If possible, factor your answer to simplest form.

Hi guys

Inverse of a Function & Restricted Domain

Inverse of a Function

When x is interchanged with y in the equation of a function y = f(x), it's reflected in the mirror line y = x. This is called an inverse function.

Find the inverse equation of f(x) = x² - 2

First, change the f(x) to y, since y = f(x).

f(x) = x² - 2
y = x² - 2

Second, switch the x variable with the y variable.

y = x² - 2
x = y² - 2

Third, isolate the y variable.

x = y² - 2
x + 2 = y²

±√ x + 2 = √y²
±√ x + 2 = y

y = ±√ x + 2

The inverse of f(x) = x² - 2 would be y = ±√ x + 2

Be careful, not all inverses are functions. If we were to perform a horizontal line test and it 'hits' two points on the graph, the inverse of the graph will not be a function.

If we were to graph our original equation, f(x) = x² - 2 its inverse would not be a function because two points hit the horizontal line test. Or if we were to graph the inverse y = f(x)-¹, it would also fail the vertical line test because there must only be one y-value for every x-value.

Tip: The inverse of multiply is divide and the inverse of addition is subtraction.

e.g., f(x) = 2x + 3

The inverse of  f(x) would be f(x)-¹ = ¹⁄₂ x - ³⁄₂

If you were given a question on how you would get the inverse to be a function then this is where restriction comes in.

Also another way to determine the domain and range of the inverse of f(x) is to switch the domain and range of f(x).
e.g., 
f(x)
D: {x | x∈R} or (-∞, ∞)
R: {y | y ≥ 0, y∈R} or [0, ∞)

f(x)-¹
D: {x | x ≥ 0, x∈R} or [0, ∞)
R: {y | y∈R} or (-∞, ∞)

Restricted Domain

Domain is the set of all possible values for the independent variable in a relation. This would be the x-values.

So now that we know that our inverse isn't a function, how would we restrict the domain so that the inverse of f(x) is a function?

Well, we could achieve this by using one side of the parabola, we could use the left side of the parabola (-∞,0] or we could use the right side of the parabola [0,∞).

If we were to use the right side of the parabola, then we would restrict our domain to [0, ∞) or { x | x ≥ 0, x∈R}.

 

We would write, "Restrict the domain of f(x) to [0, ∞)", so that the inverse will become a function.

Now if we were to graph our inverse y = ±√ x + 2 , we would use the y = √ x + 2 and not y = -√ x + 2 because we restricted our graph so that the positive values are left. Our inverse would now look like this. We could also just switch the x and y values of the restricted function and we would still get the same answer.

Now, the inverse is a function! Yay!

Now let's start from the beginning.

1. Find the inverse equation of f(x) = x² - 2

f(x) = x² - 2

y = x² - 2

x = y² - 2

x + 2 = y² 

±√ x + 2 = √y²
±√ x + 2 = y
y = ±√ x + 2

2. Restrict the domain of f(x) so that the inverse will become a function.

Restrict the domain of f(x) to [0, ∞)
y = + √ x + 2

3. List the domain and range of f(x) and its inverse.
f(x)
D: {x | x∈R} or (-∞, ∞)
R: {y | y ≥ -2, y∈R} or [-2, ∞)

f(x)-¹
D: {x | x ≥ -2, x∈R} or [-2, ∞)
R: {y | y∈R} or (-∞, ∞)

4. List the domain and range of the restricted domain of f(x) and its inverse.
f(x)
D: {x | x ≥ 0, x∈R} or [0, ∞)
R: {y | y ≥ -2, y∈R} or [-2, ∞)

f(x)-¹
D: {x | x ≥ -2, x∈R} or [-2, ∞)
R: {y | y ≥ 0, y∈R} or [0, ∞)

Stretches

VERTICAL STRETCHES

y = af(x-c)+d
In this the entire graph stretches vertically about the x- axis by a factor of "a" units.

"a" is called as vertical stretch.
So the coordinates of (x,y) translate to (x,ay).
x values does not change.

For example, y = 2f(X)+1
Hence, (x,2y) we have to multiply 2 to y coordinates and add 1.

HORIZONTAL STRETCHES

y = afb(x-c)+d
In this the entire graph stretches horizontally about y-axis by a factor of 1/b.

"b" is called as horizontal stretch.
So the coordinates of (x,y) translate to (1/bx,y). Always remember to multiply x values to reciprocal of "b".
y values does not change.

For example, y = f1/2(x)
Hence, (2x,y) as reciprocal  of 1/2 is 2. Mutilply x values to the reciprocal of 1/2.

Reflections

                                                            Reflections




      A reflection (flip) is when a point or graph is identical on either side of a line of reflection.




  reflection in the x-axis  

When reflecting about the x-axis, the y-values are affected, while the x-values stay the same

y = -f(x)

make the y-values negative - (x, y) → (x, -y)

Example:
                    What is the reflected ordered pair if (5, -2) is reflected in the x-axis?

                        - make y-values negative - (5,-2) →  (5,2) 




  reflection in the y-axis  

When reflecting about the y-axis, the x-values are affected, while the y-values stay the same

y = f(-x)

make the x-values negative - (x, y) → (-x, y)

Example:
               What is the reflected ordered pair if (1, -4) is reflected in the y-axis?

                 - make x-values negative - (1, -4) →  (-1, -4) 



  reflection in the x-axis and y-axis  

The function y = -f(-x) involves a reflection in the x-axis as well the the y-axis

make the x-values and y-values negative - (x, y) → (-x, -y)

Example: 
               What is the reflected ordered pair if (-3, 2) is reflected in both the x-axis and y-axis?

                  - make x and y values negative - (-3, 2) →  (3, -2)