Multiplying a Polynomial by a Polynomial

1. Multiplying a Binomial by a Binomial

We use the distributive law of multiplication in this case. Multiply each term of a binomial with every term of another binomial. After multiplying the polynomials we have to look for the like terms and combine them.

Example

Simplify (3a + 4b) × (2a + 3b)

Solution:

(3a + 4b) × (2a + 3b)

= 3a × (2a + 3b) + 4b × (2a + 3b)    [distributive law]

= (3a × 2a) + (3a × 3b) + (4b × 2a) + (4b × 3b)

= 6 a+ 9ab + 8ba + 12b2

= 6 a2 + 17ab + 12b2     [Since ba = ab]

2. Multiplying a Binomial by a Trinomial

In this also we have to multiply each term of the binomial with every term of trinomial.

Example

Simplify (p + q) (2p – 3q + r) – (2p – 3q) r.

Solution:

 We have a binomial (p + q) and one trinomial (2p – 3q + r)

(p + q) (2p – 3q + r)

= p(2p – 3q + r) + q (2p – 3q + r)

= 2p2 – 3pq + pr + 2pq – 3q2 + qr

= 2p2 – pq – 3q2 + qr + pr   

Therefore,

(p + q) (2p – 3q + r) – (2p – 3q) r

= 2p2 – pq – 3q2 + qr + pr – (2pr – 3qr)

= 2p2 – pq – 3q2 + qr + pr – 2pr + 3qr

= 2p2 – pq – 3q2 + (qr + 3qr) + (pr – 2pr)

= 2p2 – 3q2 – pq + 4qr – pr

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