Long and Synthetic Division

Step 1 · Divide the leading terms

x³ ÷ x = x². Write x² directly above the x³ column.

Step 2 · Multiply and subtract

Multiply x²(x−2)=x³−2x². Keep like powers in the same columns, then subtract.

Step 3 · Bring down and repeat

−2x² ÷ x = −2x. Place −2x above the x² column. Multiply, subtract, and bring down +6.

Step 4 · Finish and interpret

−3x ÷ x = −3. Place −3 above the x column, multiply, and subtract.

Quotient: x²−2x−3   Remainder: 0

Step 1 · Divide the leading terms

2x³ ÷ x = 2x². Place 2x² directly above the 2x³ column.

Step 2 · Multiply and subtract

2x²(x+3)=2x³+6x². Align the equal powers before subtracting.

Step 3 · Repeat

−3x² ÷ x = −3x. Place −3x above the x² column, multiply, subtract, and bring down −6.

Step 4 · Complete the quotient

−2x ÷ x = −2. Place −2 above the x column, multiply, and subtract.

Quotient: 2x²−3x−2   Remainder: 0

Step 1 · Insert the missing terms

The x³ term and x term are missing. Write them with zero coefficients before dividing.

x⁴+0x³+x²+0x+3

Step 2 · First division

x⁴ ÷ x=x³. Multiply x³(x−3)=x⁴−3x³ and subtract.

Step 3 · Divide the new leading term

3x³ ÷ x=3x². Multiply 3x²(x−3)=3x³−9x² and subtract.

Step 4 · Continue through the zero x term

10x² ÷ x=10x. Multiply 10x(x−3)=10x²−30x and subtract.

Step 5 · Finish

30x ÷ x=30. Multiply 30(x−3)=30x−90 and subtract.

Quotient: x³+3x²+10x+30   Remainder: 93

Step 1 · Rearrange into standard form

Arrange from the greatest exponent to the constant before dividing.

−7x⁴+5x³+x²+3x−3

Step 2 · Divide the leading terms

−7x⁴ ÷ 2x=−$\frac{7}{2}$x³. Multiply by 2x+7 and subtract.

Step 3 · Continue with the x³ term

$\frac{59}{2}$x³ ÷ 2x=+$\frac{59}{4}$x². Multiply and subtract.

Step 4 · Continue with the x² term

−$\frac{409}{4}$x² ÷ 2x=−$\frac{409}{8}$x. Multiply, subtract, and bring down −3.

Step 5 · Finish and keep the exact fractions

$\frac{2887}{8}$x ÷ 2x=+$\frac{2887}{16}$. Multiply and subtract to obtain the remainder.

Quotient: −$\frac{7}{2}$x³+$\frac{59}{4}$x²−$\frac{409}{8}$x+$\frac{2887}{16}$   Remainder: −$\frac{20257}{16}$

Step 1 · Identify c and list coefficients

x−2 means c=2. Write the coefficients 1, −4, 1, 6. Include 0 for any missing term.

Step 2 · Bring down, then multiply

Bring down 1. Multiply 1×2=2 and place 2 under −4.

Step 3 · Add and repeat

−4+2=−2. Then −2×2=−4; 1+(−4)=−3; −3×2=−6.

Step 4 · Read the quotient and remainder

6+(−6)=0. The first three bottom entries form x²−2x−3; the last entry is the remainder.

Quotient: x²−2x−3   Remainder: 0

Step 1 · Identify c and list coefficients

x−2 gives c=2. Use coefficients 2, −3, 4, −5.

Step 2 · Bring down, multiply, add

Bring down 2. Multiply 2×2=4. Add −3+4=1.

Step 3 · Continue the pattern

Multiply 1×2=2. Add 4+2=6. Multiply 6×2=12.

Step 4 · Read the final row

Add −5+12=7. The quotient coefficients are 2, 1, 6; the final entry is the remainder.

Quotient: 2x²+x+6   Remainder: 7

Step 1 · Insert the missing coefficients

The x³ term and x term are missing. Use 0 as a placeholder for each one: 1, 0, 1, 0, 3. Since x−3=x−c, use c=3.

Step 2 · Bring down and begin

Bring down 1. Multiply 1×3=3 and write 3 below the first zero. Add 0+3=3.

Step 3 · Repeat multiply, then add

Multiply 3×3=9. Add 1+9=10. Then multiply 10×3=30 and add 0+30=30.

Step 4 · Find the remainder and interpret

Multiply 30×3=90. Add 3+90=93. The first four bottom entries are the quotient coefficients; the last entry is the remainder.

Quotient: x³+3x²+10x+30   Remainder: 93

Step 1 · Rearrange and prepare the divisor

First write the dividend in standard form: −7x⁴+5x³+x²+3x−3. Ordinary synthetic division needs x−c, so factor the divisor: 2x+7=2(x+7/2)=2[x−(−7/2)]. Use c=−7/2.

Step 2 · Begin the synthetic pattern

Bring down −7. Multiply (−7)(−$\frac{7}{2}$)=$\frac{49}{2}$. Add 5+49/2=59/2.

Step 3 · Continue across every coefficient

Multiply $\frac{59}{2}$(−$\frac{7}{2}$)=−$\frac{413}{4}$ and add 1 to obtain −$\frac{409}{4}$. Then multiply again to obtain $\frac{2863}{8}$ and add 3 to obtain $\frac{2887}{8}$.

Step 4 · Find the synthetic remainder

Multiply $\frac{2887}{8}$(−$\frac{7}{2}$)=−$\frac{20209}{16}$. Add −3 to get −$\frac{20257}{16}$. This last entry is the remainder.

Step 5 · Adjust for the factor 2

The synthetic row first gives the quotient for x+7/2: −7x³+$\frac{59}{2}$x²−$\frac{409}{4}$x+$\frac{2887}{8}$. Because 2x+7 is twice x+7/2, divide every quotient coefficient by 2. The remainder does not change.