**How to divide polynomials using synthetic division:**

In order to divide polynomials using synthetic division, we must divide by a linear expression and the leading coefficient (first number) must be 1.

For example, we can use synthetic division to divide by x+3 or x – 6, but you cannot use synthetic division to divide by x^{2} + 2 or 3 x^{2} – x + 7.

Let us see some example problems to understand how to divide polynomials using synthetic division.

**Example 1 :**

Find the quotient and remainder using synthetic division

( x³ + x² - 3 x + 5 ) ÷ ( x - 1 )

**Solution :**

Let p (x) = x³ + x² - 3 x + 5 be the dividend and q (x) = x - 1 be the divisor. We shall find the quotient s(x) and the remainder r, by proceeding as follows.

q (x) = 0

x - 1 = 0

x = 1

**Step 1: **Arrange the dividend and the divisor according to the descending powers of x and then write the coefficients of dividend in the first zero. Insert 0 for missing terms.

**Step 2: **Find out the zero of the divisor.

**Step 3: **Put 0 for the first entry in the second row.

**Step 4: **Write down the quotient and remainder accordingly. All the entries except the last one in the third row constitute the coefficients of the quotient.

When P (x) is divided by (x - 1), the quotient is x² + 2 x - 1 and the remainder is 4.

Quotient = x² + 2 x - 1

Remainder = 4

**Example 2 :**

Find the quotient and remainder using synthetic division

( 3 x³ - 2 x² + 7 x - 5 ) ÷ ( x + 3 )

**Solution :**

When P (x) is divided by (x + 3), the quotient is 3 x² - 11 x + 40 and the remainder is - 125.

Quotient = 3 x² - 11 x + 40

Remainder = - 125

**Example 3 :**

Find the quotient and remainder using synthetic division

( 3 x³ + 4 x² - 10 x + 6 ) ÷ ( 3 x - 2 )

**Solution :**

Let p (x) = 3 x³ + 4 x² - 10 x + 6 be the dividend and q (x) = 3 x - 2 be the divisor. We shall find the quotient s(x) and the remainder r, by proceeding as follows.

q (x) = 0

3 x - 2 = 0

3 x = 2

x = 2/3

When P (x) is divided by (x - 1), the quotient is 3 x² + 6 x - 6 and the remainder is 2.

3 x² + 6 x - 6

Dividing the whole equation by 3,we get

Quotient = x² + 2 x - 2

Remainder = 4

**Example 4 :**

Find the quotient and remainder using synthetic division

( 3 x³ - 4 x² - 5 ) ÷ ( 3 x + 1 )

**Solution :**

Let p (x) = 3 x³ - 4 x² - 5 be the dividend and q (x) = 3 x + 1 be the divisor. We shall find the quotient s(x) and the remainder r, by proceeding as follows.

q (x) = 0

3 x + 1 = 0

3 x = - 1

x = -1/3

When P (x) is divided by (3x + 1), the quotient is 3 x² - 5 x + 5/3 and the remainder is -50/9.

Quotient = 3 x² - 5 x + 5/3

Remainder = -50/9

**Example 5 :**

Find the quotient and remainder using synthetic division

( 8 x⁴ - 2 x² + 6 x - 5 ) ÷ ( 4 x + 1 )

**Solution :**

Let p (x) = 8 x⁴ - 2 x² + 6 x - 5 be the dividend and q (x) = 4 x + 1 be the divisor. We shall find the quotient s(x) and the remainder r, by proceeding as follows.

q (x) = 0

4 x + 1 = 0

4 x = -1

x = -1/4

When P (x) is divided by (4 x + 1), the quotient is 8 x³ - 2 x² - (3/2) x + (51/8) and the remainder is -211/32.

Quotient = 8 x³ - 2 x² - (3/2) x + (51/8)

Remainder = -211/32

After having gone through the stuff given above, we hope that the students would have understood "How to divide polynomials using synthetic division".

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