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Dividing a Trinomial by a Monomial

To divide a trinomial by a monomial, we should follow these steps:

Step 1: Divide each term of the trinomial by the monomial by putting each term of trinomial in
the numerator and the monomial in the denominator separately.

Step 2: Divide the coefficients and use the exponent rules to simplify the variables in each term.

Step 3: Write all the quotients obtained and then add to get the final quotient.
2
3
Example 16: Divide (ax – bx + cx) by (–dx).
2
3
ax − bx + cx
3
Solution: We have (ax – bx + cx) ÷ (–dx) =
2
− ( dx)
ax 3 bx 2 cx a b c
= − + =− x 31 + x 21 − x 11
−
−
−
(− dx) (− dx) (− dx) d d d
a b c
0
= − x 2 + x − (Q x = 1)
d d d
Dividing a Polynomial by a binomial or a polynomial
To divide a polynomial by a binomial or a polynomial, we use the long division method:
Step 1: Arrange the terms of the dividend and the divisor in descending order based on their
exponents.

Step 2: Take the first term of the dividend and divide it by the first term of the divisor.
Step 3: Multiply the divisor by the resultant term of quotient we got in step 2, i.e., the first term of
the quotient and place this product below the dividend. So that like terms are aligned.
Step 4: Subtract the product from the dividend. If there are more terms in the dividend bring down
the next one to form a new dividend.

Step 5: Keep repeating the above process until we end up with a remainder, which can either be
zero or a polynomial of a degree less than that of the divisor.
3
2
Example 17: Divide (18a – 6a + 5a – 110) by (a + 2).
Solution: First, arrange the terms of the dividend and the divisor in descending order of their
2
3
exponents. That is, –6a + 18a + 5a – 110.
3
2
Now, put the dividend, –6a + 18a + 5a – 110, inside the long division bar and put the divisor,
a + 2 outside as follows:
2
a + 2 –6a + 18a + 5a – 110
3
−6a 3
2
Divide the first term of the dividend by the first term of the divisor, =−6a .
a
Note: If the first term of the dividend and the divisor have the same sign, the quotient obtained has the
positive sign. If the first term of the dividend and the divisor have opposite signs, the quotient obtained
has the negative sign.

Now, multiply the divisor a + 2 by the term (–6a ), and place the product –6a – 12a below the
2
2
3
dividend.
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