#### notes

**Multiplying a Binomial by a Trinomial:**

- In this multiplication, we shall have to multiply each of the three terms in the trinomial by each of the two terms in the binomial.

**1) (a + 7) × (a ^{2} + 3a + 5) **

= a × (a

^{2}+ 3a + 5) × 7 × (a

^{2}+ 3a + 5) .... [using the distributive law]

= a

^{3}+ 3a

^{2}+ 5a + 7a

^{2}+ 21a + 35

= a

^{3}+ (3a

^{2}+ 7a

^{2}) + (5a + 21a) + 35

= a

^{3}+ 10a

^{2}+ 26a + 35

**2) Simplify (a + b)(2a – 3b + c) - (2a – 3b)c.**

⇒ We have (a + b)(2a – 3b + c) = a (2a – 3b + c) + b (2a – 3b + c)

= 2a^{2} – 3ab + ac + 2ab – 3b^{2} + bc

= 2a^{2} – ab – 3b^{2} + bc + ac

and (2a – 3b)c = 2ac – 3bc

Therefore,

(a + b)(2a – 3b + c) – (2a – 3b)c

= 2a^{2} – ab – 3b^{2} + bc + ac – (2ac – 3bc)

= 2a^{2} – ab – 3b^{2} + bc + ac – 2ac + 3bc

= 2a^{2} – ab – 3b^{2} + (bc + 3bc) + (ac – 2ac)

= 2a^{2} – 3b^{2} – ab + 4bc – ac.

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