LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

MULTIPLICATION OF ALGEBRAIC EXPRESSION OF BINOMIAL BY TRINOMIAL

__MULTIPLICATION OF ALGEBRAIC EXPRESSION OF BINOMIAL BY TRINOMIAL -__

**Multiplying an algebraic expression of a trinomial by a binomial is a common operation in algebra. To do this, you can use the distributive property (also known as the FOIL method when multiplying a binomial by a binomial). Here's how you can multiply a trinomial by a binomial step by step:**

**Let's say you have a trinomial, such as ax²+ bx + c, and a binomial, such as dx + e, and you want to multiply them together.**

**Start by multiplying the first term of the trinomial (ax²) by each term of the binomial (dx and e):**

** ax²⋅dx = adx³ **

** ax²⋅e = aex² **

** 2. Next, multiply the second term of the trinomial (bx) by each term of the binomial:-**

** 2bx⋅dx = bdx² **

** bx⋅e = bex**

** 3. Finally, multiply the third term of the trinomial (c) by each term of the binomial:-**

** c⋅dx = cdx**

** c⋅e = ce**

** 4. Now, add up all the products you obtained in steps 1, 2, and 3 to get the final result:**

** adx³+ aex²+ bdx²+ bex + cdx + ce**

**This is the product of the trinomial ax²+ bx + c and the binomial dx + e.**

**You can simplify the expression further by combining like terms if necessary.**

**Keep in mind that this process follows the distributive property, which states that for any real numbers or algebraic expressions a, b, c, d, and e:**

** (a+b) ⋅ (c+d+e) = ac + ad + ae + bc + bd + be**

**So, when you multiply a trinomial by a binomial, you apply this property repeatedly to each term in the trinomial and each term in the binomial.**

__ANOTHER WAY OF UNDERSTANDING -__

**Multiplying an algebraic expression of a trinomial by a binomial involves applying the distributive property of multiplication over addition (or subtraction). Here's how you can do it step by step:**

**Let's say you have a trinomial, such as ax²+ bx + c, and a binomial, such as dx + e. To multiply the trinomial by the binomial, follow these steps:**

** 1. Distribute the first term of the binomial (dx) to each term in the trinomial:-**

** (ax²+ bx + c) (dx) = ax²(dx) + bx (dx) + c (dx)**

** 2. Distribute the second term of the binomial (e) to each term in the trinomial:- **

** (ax²+ bx + c) (e) = ax²(e) + bx (e) + c (e)**

** 3. Now, you have two sets of terms from the distribution in step 1 and step 2:**

** Set 1:- ax²(dx), bx (dx), c (dx)**

** Set 2:- ax²(e), bx (e), c (e)**

** 4. Finally, combine the like terms in each set:-**

** Set 1:- ax² (dx) + bx (dx) + c (dx) = (adx²) + (bdx) + (cdx)**

** Set 2:- ax²(e) + bx (e) + c(e) = (aex²) + (be) + (ce)**

**Now, you have two simplified expressions:**

** (ax²+ bx +c) (dx + e) = (adx²+ bdx + cdx) + (aex²+ be + ce)**

**You can further simplify or combine like terms if needed. This is the result of multiplying an algebraic expression of a trinomial by a binomial using the distributive property.**

**Example.1) Multiplication of (4x + 2) and (x + y + z) will be**

** (4x²+ 4xy + 4xz + 2x + 2y +2z).**

**Example.2) Multiplication of (2x²+ 2xy) and (2x + y + z) will be**

**Ans.) (4 x 3 + 6x²y + 2x²z + 2xy² + 2xyz).**

__Examples based on Multiplying a Polynomial by a Polynomial:-__

**Example.1) Multiply the binomials (2ab +3b²) and (3ab – 2b²).**

**Ans.) **

**(2ab +3b²) x (3ab – 2b² ) **

**= 2ab x (3ab
– 2b²) + 3b² x (3ab – 2b²)**

**= 6a²b² – 4ab³ + 9ab³– 6b⁴**

**= 6a²b² + 5ab³ –
6b⁴ (Ans.)**

**Example.2) Simplify (a + b + c) (a + b – c)**

**Ans.)**

**(a + b +c) (a + b – c) **

**= a (a + b – c) + b (a + b
– c) + c (a + b – c)**

**= a²+ ab
– ac + ab + b² – bc + ac + bc – c²**

**= a²+ b² – c² + 2ab (Ans.)**