1)let f(x) be a polynomial function explian how to use the factor theorem to check if (x-c) is a factor of F(x)
2) use synthetic division to factor X^2-2x^2-9x+18 completly
if f(c) = 0 , then x-c is a factor
for f(x) = x^2 - 2x^2 - 9x + 18
why do you have two x^2 terms, I will assume the first is x^3
If so, we don't need the factor theorem for this one, grouping is obvious
x^3 - 2x^2 - 9x + 18
= x^2(x-2) - 9(x-2)
= (x-2)(x^2 - 9)
= (x-2)(x+3)(x-3)
if you had not seen this, try x = ±1, ±2 , ±3 , that is factors of 18
f(1) ≠ 0
f(-1) ≠ 0
f(2) = 0 , yeahhhh, so x-2 is a factor
..
f(3) = 0 , so x-3 is a factor
f(-3) = 0 , so x+3 is a factor
since we have a cubic, there can only be a maximum of 3 algebraic factors
so as above
(x-2)(x+3)(x-3)
1) To use the Factor Theorem to check if (x-c) is a factor of F(x), you need to perform synthetic division. Here's a step-by-step explanation of how to do it:
Step 1: Write down the polynomial function F(x) in the form of a coefficient list, with the powers of x descending from left to right. For example, if F(x) = ax^3 + bx^2 + cx + d, the coefficient list would be [a, b, c, d].
Step 2: Set up the synthetic division table. Write down the value of c (the possible root) on the left-hand side of the table, and then list the coefficients of F(x) in the top row of the table.
Step 3: Begin the synthetic division process. Starting with the first coefficient in the top row (the coefficient of the highest degree term), follow these steps:
a) Bring down the first coefficient below the line.
b) Multiply the value you brought down by the value of c, then write the result below the next coefficient in the top row.
c) Add the two values together, and write the sum below the line.
d) Repeat the process for the remaining coefficients, until you reach the last coefficient.
Step 4: Check the final value on the bottom row of the synthetic division. If the result is zero, then (x-c) is a factor of F(x).
2) Let's use synthetic division to completely factorize the polynomial F(x) = x^2 - 2x^2 - 9x + 18.
Step 1: First, write down the coefficient list of the polynomial: [-2, -9, 18].
Step 2: Set up the synthetic division table. Since we are looking for factors, we need a possible root. Let's try c = 1, so write down '1' on the left side of the table, and list the coefficients of F(x) in the top row: [1, -2, -9, 18].
| 1 -2 -9 18
1 |________________
|
Step 3: Begin the synthetic division process by following the steps for each coefficient:
a) Bring down the first coefficient (-2) below the line.
b) Multiply the value you brought down (-2) by the value of c (1), then write the result (-2) below the next coefficient (-2) in the top row.
c) Add the two values together (-2 + -2), and write the sum (-4) below the line.
d) Repeat the process for the remaining coefficients:
| 1 -2 -9 18
1 |________________
-2 0 -9
Step 4: Check the final value on the bottom row (-9). Since it is not equal to zero, (x-1) is not a factor of F(x).
Therefore, we need to try another possible root in order to continue factoring the polynomial completely.