If f(x)=2x^4-9x^3+6x-5, then use synthetic division to find f(6)
What is synthetic division? How do I solve this?
6 │ 2 -9 0 6 -5
...--------------
......12 18 108684
....2 3 18 114 679
It is so hard to line things up properly,
in the last two rows, the 3 is under the 12, 18's line up, 108 and 104 line up.
the answer is the last number, namely 679
Synthetic division is a method used to divide a polynomial by a linear factor (x - c), where c is a constant. It allows you to find the remainder of the division quickly. Here's how you can solve the problem using synthetic division:
Step 1: Write down the coefficients of the polynomial in descending order, including any missing terms. In this case, the polynomial is f(x) = 2x^4 - 9x^3 + 6x - 5. So, the coefficients are 2, -9, 0, 6, -5.
Step 2: Determine the value of c, which represents the number you want to substitute into the polynomial. In this case, we want to find f(6), so c is 6.
Step 3: Set up the synthetic division. Write down the coefficients from step 1 and then write c outside the division symbol.
6 | 2 -9 0 6 -5
Step 4: Perform the synthetic division. Bring down the first coefficient (2) and multiply it by c (6) to get 12. Write the result below the second coefficient (-9). Add the result with the second coefficient to get 3. Multiply 3 by c (6) to get 18, and write the result below the third coefficient (0). Repeat this process until you reach the last coefficient.
6 | 2 -9 0 6 -5
| 12 18 108 672
------------------------
2 3 18 114 667
Step 5: Interpret the result. The last value in the bottom row is the remainder, and the other numbers represent the coefficients of the quotient. In this case, the remainder is 667.
Step 6: Find f(6). Since the remainder is zero when using synthetic division, f(6) = 0.
Therefore, f(6) = 0.
Synthetic division is a method used to divide a polynomial by a linear binomial of the form (x - a), where "a" is a constant. It simplifies the long division process by using coefficients of the polynomial.
To solve the given problem using synthetic division, follow these steps:
Step 1: Set up the synthetic division using the given linear binomial and the coefficients of the polynomial. In this case, the linear binomial is (x - 6) and the coefficients of the polynomial are 2, -9, 6, and -5. The coefficients are written in descending order of powers of x.
6 | 2 -9 6 -5
Step 2: Bring down the first coefficient (2) below the division line.
6 | 2
----
Step 3: Multiply the divisor (6) by the number below the division line (2) and write the result below the next coefficient. In this case, 6 * 2 = 12.
6 | 2
12
----
Step 4: Add the result to the next coefficient. In this case, 12 + (-9) = 3.
6 | 2
12
----
3
Step 5: Repeat steps 3 and 4 until all coefficients have been processed.
6 | 2 -9 6
12 18
------
3 6
Step 6: The number at the bottom right corner is the remainder.
6 | 2 -9 6 -5
12 18 24
------------
3 6 19
Step 7: The coefficients on the top row of the synthetic division represent the quotient of the division. In this case, the quotient is 2x² - 9x + 6.
Therefore, using synthetic division, we have f(6) = 2(6)² - 9(6) + 6 = 72 - 54 + 6 = 24.
Hmmmm. I don't see how synthetic division can help on this. f(6) is just plug in 6 for x and grind.
Synthetic division helps dividing polynomials, which is not here .