use synthetic division to factor x^3-2x^2-9x+18
if you know how to do synthetic division, it should be easy to show that
x^3-2x^2-9x+18 = (x+3)(x-2)(x-3)
you know that all the roots have to be factors of 18.
x^3-2x^2-9x+18 = (x+3)(x-2)(x-6)
To factor the polynomial x^3 - 2x^2 - 9x + 18 using synthetic division, follow these steps:
Step 1: Determine the possible rational roots:
The possible rational roots of a polynomial are the numbers that can be formed by taking the factors of the constant term (in this case, 18) and dividing them by the factors of the leading coefficient (in this case, 1).
Factors of 18: ±1, ±2, ±3, ±6, ±9, ±18
Factors of 1: ±1
Possible rational roots: ±1, ±2, ±3, ±6, ±9, ±18
Step 2: Find the actual rational roots:
To find the rational roots, substitute each possible rational root into the polynomial and check if the result is zero.
Substituting x = 1:
(1)^3 - 2(1)^2 - 9(1) + 18 = 1 - 2 - 9 + 18 = 8 (not zero)
Substituting x = -1:
(-1)^3 - 2(-1)^2 - 9(-1) + 18 = -1 - 2 + 9 + 18 = 24 (not zero)
Substituting x = 2:
(2)^3 - 2(2)^2 - 9(2) + 18 = 8 - 8 - 18 + 18 = 0 (zero)
x = 2 is a root.
Step 3: Perform synthetic division:
Divide the polynomial by (x - 2) using synthetic division.
2 | 1 -2 -9 18
|_______2 0 -18
____________________
1 0 -9 0
The result of the synthetic division is 1x^2 + 0x - 9, with no remainder.
Step 4: Write the factored form:
The factored form of the polynomial x^3 - 2x^2 - 9x + 18 is:
(x - 2)(x^2 - 9)
Now, we can further factor the quadratic term. The factorization of x^2 - 9 is (x - 3)(x + 3).
Therefore, the complete factored form of the polynomial is:
(x - 2)(x - 3)(x + 3)
To factor the polynomial x^3 - 2x^2 - 9x + 18 using synthetic division, follow these steps:
Step 1: Make sure the polynomial is written in descending order of powers.
The given polynomial, x^3 - 2x^2 - 9x + 18, is already in descending order.
Step 2: Determine the possible rational roots (factors of the constant term divided by factors of the leading coefficient).
The constant term is 18, and the leading coefficient is 1. Possible rational roots are the divisors of 18 divided by the divisors of 1. Therefore, the possible rational roots are ±1, ±2, ±3, ±6, ±9, and ±18.
Step 3: Use synthetic division to test the possible rational roots.
Let's start by testing the root x = 1.
1 | 1 - 2 - 9 + 18
Perform the synthetic division:
1
-------------
1 - 1 | -1 -5 -4
-1 0 -4
-------------
0 -5 -8
The coefficients in the bottom row represent the quotient of the division. In this case, the quotient is x^2 - 5x - 8.
Step 4: Repeat the synthetic division process.
Now, use the quotient obtained in step 3, which is x^2 - 5x - 8, and follow the steps again to find the remaining factors.
1 | 0 - 5 - 8
Perform the synthetic division:
1
------------
1 - 5 | 0 -5
0 0
------------
0 -5
The coefficients in the bottom row represent the quotient of the division. In this case, the quotient is x - 5.
Step 5: Write the factored form of the polynomial.
The factored form of the polynomial x^3 - 2x^2 - 9x + 18, using synthetic division, is:
(x - 1)(x - 5)(x + 1)
Therefore, the factored form of x^3 - 2x^2 - 9x + 18 is (x - 1)(x - 5)(x + 1).