Question

Dividing polynomials using synthetic and long division:

For #64-66, a polynomial and a factor are given. Find the quotient and show all work.

64. (x⁴-3x³+6x²-6x+8) ; (x²+2)
65. (4x³+x+15); (2x+3)
66. (2x³-15x²+29x-4); (x-4)​

Answer 1

64. x^2 -3x +4

65. 2x^2 -3x +5

66. 2x^2 -7x +1

Explanation:

The work is shown in the attachments. The first three attachments show the polynomial long division. The last attachment shows the synthetic division for all three problems.

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Problem 64 is unique in that the synthetic division requires the product of the divisor and the quotient term to be added two columns to the right instead of the usual next column. (The spreadsheet formula was adjusted accordingly.)

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Problem 65 requires the dividend and divisor both be divided by the divisor's leading coefficient, so that coefficient becomes 1. This gives numbers with fractions, but the arithmetic is carried out in the usual way. Doing this division up front means no adjustment is needed in the quotient.

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Comment on synthetic division

Synthetic division works basically the same way that polynomial long division works, except the leading term of the divisor is missing from all calculations. Since the point of the division process is to bring any sums involving that term to zero, nothing is lost by leaving out a bunch of zeros.

The other difference is that the divisor is negated to start with, so that addition is performed at each stage, rather than the subtraction you see in long division. Addition is generally easier to do without error, so that's a nice improvement over long division, too.

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It turns out to be really handy to do synthetic division on a spreadsheet. The formulas are easily made and copied, and the math is subsequently done by the spreadsheet without error.