Use synthetic division to divide the polynomial 2x^3 – 12x – 5 by x + 4, and write the quotient polynomial and the remainder. [Be careful – notice that there is no x^2 term.] Please Show work.
-4 │ 2 0 -12 -5
.......-8 32 -80
2 -8 20 -85
so (2x^3 – 12x – 5) ÷ (x + 4)
= 2x^2 - 8x + 20 - 85/(x+4)
To perform synthetic division, we need to follow these steps:
Step 1: Write the polynomial in descending order with missing terms represented by zeros. In this case, the polynomial is already in descending order:
2x^3 - 12x + 0x - 5
Step 2: Identify the divisor and set it equal to zero to solve for x:
x + 4 = 0
x = -4
Step 3: Set up the synthetic division table. Write the coefficient of each term of the dividend (2x^3 - 12x + 0x - 5) in a row, excluding the exponents, and then write the value of x that we obtained from step 2 (in this case, -4) on the left side of the table.
-4 | 2 0 -12 -5
Step 4: Bring the leading coefficient (2) straight down below the horizontal line:
2
-4 | 2 0 -12 -5
Step 5: Multiply the value just brought down by the divisor (-4), and write the result in the next column:
2
-8
-4 | 2 0 -12 -5
Step 6: Add the result obtained in step 5 to the next coefficient:
2
-8 | 0
-4 | 2 -8 -12 -5
Step 7: Repeat steps 5 and 6 for each coefficient until all coefficients have been used:
2
-8 | 0 -32
-4 | 2 -8 -12 -5
Step 8: The number at the bottom-right of the table represents the remainder (-32), while the other values in the bottom row represent the coefficients of the quotient polynomial:
Remainder: -32
Quotient Polynomial: 2x^2 - 8x - 12
Therefore, the quotient polynomial is 2x^2 - 8x - 12, and the remainder is -32.