Consider the function f(x) = x4 – 3x3 – 7x2 + 15x + 18.
Use synthetic division to divide f(x) by x – 3. Use the answer to explain whether or not x – 3 is a factor of f(x).Factor f(x) completely.
take a visit to
http://www.mathportal.org/calculators/polynomials-solvers/synthetic-division-calculator.php
and enter your coefficients. It will show the calculation, and even explain it if needed.
It's hard to do it here, due to browser formatting of html.
thank you, Steve :)
To use synthetic division to divide f(x) by x - 3, we will set up the synthetic division table.
| 3 | -3 | -7 | 15 | 18 |
___________________________________________________
3 | 1 | -3 | -7 | 15 | 18
___
3 | 0 | -9 | -24 | 33
___
1 | -3 | -16 | -9
The numbers on the left side of the division bar represent the coefficients of the function f(x), while the number on the top right represents the divisor, x - 3.
The last number on the bottom row, 1, is the remainder. The other numbers on the bottom row represent the coefficients of the quotient.
Therefore, the quotient is x^3 - 3x^2 - 16x - 9, and the remainder is 1.
Since the remainder is non-zero (1 in this case), x - 3 is not a factor of f(x).
To factor f(x) completely, we can use the information from the quotient obtained in the synthetic division:
f(x) = (x - 3)(x^3 - 3x^2 - 16x - 9)
To factor the remaining cubic polynomial, we can use any applicable factoring techniques such as grouping, factoring by grouping, or the cubic formula.
To use synthetic division to divide the polynomial f(x) by x - 3, we follow these steps:
Step 1: Write down the coefficients of f(x) in descending order:
f(x) = x^4 - 3x^3 - 7x^2 + 15x + 18
Step 2: Set up the synthetic division table. Write the divisor (x - 3) to the left and the coefficients of f(x) to the right:
3 | 1 -3 -7 15 18
|_________________
Step 3: Bring down the first coefficient, which is 1, and write it under the horizontal line:
3 | 1 -3 -7 15 18
|_________________
| 1
Step 4: Multiply the divisor, 3, by the number you just brought down, 1. Write the result of the multiplication below and add it to the second coefficient:
3 | 1 -3 -7 15 18
| 3
|__________
| 1 0
Step 5: Repeat this process for the following coefficients. Multiply 3 by 0 and add it to -3:
3 | 1 -3 -7 15 18
| 3 6
|__________
| 1 0 -7
Step 6: Continue multiplying and adding until you reach the final coefficient:
3 | 1 -3 -7 15 18
| 3 6 -3
|__________
| 1 0 -7 12
Step 7: The last number obtained, 12, represents the remainder. If the remainder is zero, it means that x - 3 is a factor of f(x). In this case, the remainder is not zero. Therefore, x - 3 is not a factor of f(x).
To factor f(x) completely, we need to find the roots of the polynomial. Since x - 3 is not a factor, we can look for other possible factors. One way to do this is by using the Rational Root Theorem.
According to the Rational Root Theorem, the possible rational roots of f(x) are the factors of the constant term (18) divided by the factors of the leading coefficient (1). In this case, the factors of 18 are ±1, ±2, ±3, ±6, ±9, and ±18, and the factors of 1 are ±1. Therefore, the possible rational roots are ±1, ±2, ±3, ±6, ±9, and ±18.
We can try these values using synthetic division to check for any roots. If a value gives a remainder of 0, it means it is a root/factor of f(x). If not, we move on to the next value until we find a root.
Let's try x = 1:
1 | 1 -3 -7 15 18
|_________________
| 1
1 | 1 -3 -7 15 18
| 1
|__________
| 1 -2
Since the remainder is not 0, x = 1 is not a root/factor.
We can try the remaining possible roots in a similar manner until we find a root. In this case, x = -3 is a root. Hence, we can factor f(x) accordingly:
f(x) = (x + 3)(x^3 - 6x^2 + 5x + 6)
To factor the remaining cubic polynomial, you can continue to use synthetic division or other factoring methods.