Create a quadratic polynomial function f(x) and a linear binomial in the form (x − a).

Question

Create a quadratic polynomial function f(x) and a linear binomial in the form (x − a).

Part 1. Show all work using long division to divide your polynomial by the binomial.
Part 2. Show all work to evaluate f(a) using the function you created.
Part 3. Use complete sentences to explain how the remainder theorem is used to determine whether your linear binomial is a factor of your polynomial function

Answer 1

I do not understand this question.

Answer 2

suppose we have

f(x) = 3x^3 - 5x^2 + 4x + 1
and we divide that by x-2
( I can't show the long algebraic division in this format, but I am sure you know how to do that)

you should have 3x^2 + x + 6 with a remainder of 13

now let's find f(2)
f(2) = 3(8) - 5(4) + 4(2) + 1
= 24 - 20 + 8 + 1
= 13

What would happen if the remainder is zero?
Wouldn't your original expression factor?

So as long as f(a) gives you a non-zero result, you know that the expression would NOT factor,
conversely .....

Answer 3

Part 1:

Let's create the quadratic polynomial function f(x) = x^2 + 3x + 2 and the linear binomial (x - a).

To divide the polynomial f(x) by the binomial (x - a), we can use long division. Here's how we do it:

Step 1: Write the polynomial and the binomial in the division format:
x^2 + 3x + 2 ÷ (x - a)

Step 2: Divide the first term of the polynomial by the first term of the binomial:
x^2 / x = x

Step 3: Multiply the quotient from step 2 by the binomial:
x * (x - a) = x^2 - ax.

Step 4: Subtract the result from step 3 from the original polynomial:
(x^2 + 3x + 2) - (x^2 - ax) = x^2 + 3x + 2 - x^2 + ax = (3x + 2 + ax)

Step 5: Repeat steps 2-4 with the new polynomial (3x + 2 + ax):
First term: 3x / x = 3
New polynomial: (3x + 2 + ax) - (3(x - a)) = 3x + 2 + ax - 3x + 3a = (2 + 3a)

Part 2:

To evaluate f(a) using the function f(x) = x^2 + 3x + 2, we substitute x with the value a in the function:
f(a) = a^2 + 3a + 2

Part 3:

The remainder theorem states that if a polynomial f(x) is divided by a linear binomial (x - a) and the remainder is zero, then (x - a) is a factor of f(x).

In our case, after performing long division, we obtained a remainder of (2 + 3a). Since the remainder is not zero, we can conclude that the linear binomial (x - a) is not a factor of the polynomial x^2 + 3x + 2.

Answer 4

What have you got so far for this assignment?

You didn't expect me to just do it for you, did you?