Use synthetic division and the given factor to completely factor the polynomial.
x^3-x^2-24x-36; (x-6)
How do I do this?
(x^3-x^2-24x-36)/(x-6)= ( x^2 +5x +6) = ( x +3)( x +2)
I think?
Researching would do me no good. The last math class I had was as a high school junior about 65 years ago.
Wait for a real math tutor to help you.
ok thanks anyways
Can you try to research it really quick? I have tried and found nothing.
Well, let's start with some mathematics and add a little humor to make it fun! To completely factor the polynomial using synthetic division, we first set up our synthetic division table with the divisor (x-6) on the outside, and write the coefficients of our polynomial inside the box.
6 | 1 -1 -24 -36
To use synthetic division, we bring down the first coefficient (1) and then multiply it by the divisor (6). That gives us 6, which we then write below the next coefficient (-1). We then add these two numbers together to get 5. Now, we multiply this sum (5) by the divisor (6), which gives us 30. Write down 30 below -24, and add them together to get 6.
6 | 1 -1 -24 -36
----------------------
1 5 6 -30
We repeat the process with this new number (6), multiplying it by the divisor (6) and writing the result (-36) below the last coefficient. When we add them together, we get 0. Since we have no remainder, this means that (x-6) is indeed a factor of our polynomial, and we can now write it in factored form.
The factored form of the polynomial x^3-x^2-24x-36 is:
(x-6)(x^2+5x+6)
Voila! We've factored the polynomial while keeping the funny bone intact. Keep smiling and solving!
To use synthetic division and the given factor (x-6) to completely factor the polynomial x^3-x^2-24x-36, follow these steps:
Step 1: Set up the synthetic division table by placing the factor (x-6) on the outside of the box and writing the coefficients of the polynomial inside the box.
6 | 1 -1 -24 -36
Step 2: Bring down the first coefficient, which is 1, into the leftmost box of the bottom row.
6 | 1 -1 -24 -36
| 1
Step 3: Multiply the factor (6) by the number in the bottom row and write the result in the second box of the bottom row.
6 | 1 -1 -24 -36
| 1 6
Step 4: Add the number in the second box of the bottom row to the coefficient in the next column of the polynomial and write the result in the third box of the bottom row.
6 | 1 -1 -24 -36
| 1 6 30
Step 5: Continue this process until you reach the end of the polynomial.
6 | 1 -1 -24 -36
| 1 6 30 6
Step 6: The numbers in the bottom row represent the coefficients of the resulting polynomial. In this case, we obtained a quotient of 1x^2 + 6x + 30 and a remainder of 6.
Step 7: Write the completely factored form using the given factor and the quotient obtained from synthetic division.
x^3 - x^2 - 24x - 36 = (x - 6)(x^2 + 6x + 30) + 6
Therefore, the completely factored form of the polynomial is (x - 6)(x^2 + 6x + 30) + 6.