Multiplying Polynomials

An algebraic expression is a mathematical phrase that contains integral or fractional constants (numbers), variables (alphabets) and algebraic operators (such as add, subtract, divide, multiply, etc.) operating on them.

Algebraic expression with one or more terms together is known as polynomial.

Multiplying Polynomials:

(i) Binomial by a Binomial-

When we multiply a binomial with a binomial, we follow the Distributive law of multiplication.

We know that Binomial have 2 terms. Multiplying two binomials give the result having maximum of 4 terms (only in case when we don’t have like terms). In case of like terms the total number of terms is reduced.

Like Terms-

According to Commutative Law of Multiplication, the terms like ab and ba gives the same result. Thus they can be written in both the forms.

Eg. \(5 \times 6 = 6 \times 5 = 30\)

Now, Consider two binomials given as  (a+b) and (m+n).

Multiplying them we have,

\((a+b) \times (m+n)\)

\(\Rightarrow a \times (m+n) + b \times (m+n)\)               (Distributive law of multiplication)

\(\Rightarrow (am + an) + (bm + bn)\)                                       (Distributive law of multiplication)

Thus \((a+b) \times (m+n) = am + an + bm + bn\).

Example- Find the result of multiplication of two polynomials (6x +3y) and (2x+ 5y).

Solution- \((6x-3y) \times (2x+5y)\)

\(\Rightarrow 6x \times (2x+5y) – 3y \times (2x+5y)\)               (Distributive law of multiplication)

\(\Rightarrow (12x^{2} + 30xy) – (6yx + 15y^{2})\)                    (Distributive law of multiplication)

\(\Rightarrow \; 12x^{2} + 30xy – 6xy – 15y^{2}\)                     (as xy = yx)

Thus \((6x+3y) \times (2x+5y) = 12x^{2} + 24xy -15y^{2} \)..

Let us take up an example. Say, you are required to multiply a binomial (5y + 3z) with another binomial (7y – 15z). Let us see how it is done.

(5y + 3z) x (7y – 15z)

= 5y x (7y – 15z) + 3z x (7y – 15z)                               (Distributive law of multiplication)

= (5y x 7y) – (5y x 15z) + (3z x 7y) – (3z x 15z)       (Distributive law of multiplication)

= 35y2 – 75yz + 21zy – 45z2

= 35y2 – 75yz + 21yz – 45z2                                           (yz = zy)

= 35y2 –54yz – 45z2

Multiplying Binomial with a Trinomial-

When multiplying polynomials, that is, a binomial by a trinomial, we follow the distributive law of multiplication. Thus, 2 x 3 = 6 terms are expected to be in the product. Let us take up an example.

(a2 – 2a) x (a + 2b – 3c)

= a2 x (a + 2b – 3c) – 2a x (a + 2b – 3c)                                                      (Distributive law of multiplication)

= (a2x a) + (a2 x 2b) + (a2 x -3c) – (2a x a) – (2a x 2b) – (2a x -3c)   (Distributive law of multiplication)

= a3 + 2a2b – 3a2c – 2a2 – 4ab + 6ac

= a3 – 2a2 + 2a2b – 3a2c– 4ab + 6ac                                                            (Rearranging the terms)

Thus, when we are multiplying polynomials, we need to keep the following pointers in mind:

  • Distributive Law of multiplication is used twice when 2 polynomials are multiplied.
  • Look for the like terms and combine them. This may reduce the expected number of terms in the product.
  • Preferably, write the terms in the decreasing order of their exponent.
  • Be very careful with the signs when you open the brackets.