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Multiplying Binomial
A binomial is defined as an algebraic expression that has two terms connected by a plus or a minus sign. Multiplying binomials is similar to the multiplication of two whole numbers or fractions. We will be learning about different methods to understand the concept of multiplying binomials.
How to Multiply Binomials?
Multiplying binomials is similar to the multiplication of any 2 digit numbers. The only difference is that it uses the concept of multiplication of algebraic expressions. The terms of one binomial are multiplied by the terms of the other binomial. After this step, the algebraic sum of these products is taken. Let us learn about the different methods that are used to multiply binomials.
Multiplying Binomials using Distributive Property
One of the methods for multiplying binomials is using the distributive property of multiplication twice. Let's take two binomials (x + 2) and (x + 3) and multiply them with the help of the following steps.
 Step 1: To multiply (x + 2)(x + 3), we will take the first term of the first binomial and multiply it with the second binomial, i.e., x(x + 3)
 Step 2: Now, we will take the second term of the first binomial and multiply it with the second binomial, i.e., 2(x + 3)
 Step 3: We will combine the results of Step 1 and Step 2 and add them, i.e., x(x + 3) + 2(x + 3)
 Step 4: Now we will apply the distributive property to x(x + 3) and 2(x + 3) and individually expand them, i.e., x(x + 3) = x^{2} + 3x and 2(x + 3) = 2x + 6
 Step 5: We will now add the results obtained in Step 4 by combining the like terms, i.e., x^{2} + 3x + 2x + 6 = x^{2} + 5x + 6
Thus, the product of (x + 2)(x + 3) = x^{2} + 5x + 6. Let us understand this with the help of the calculation shown below.
NOTE: This method is also known as the Horizontal Distributive Method. This method can be applied to any polynomial multiplication.
Multiplying Binomials Using the FOIL Method
The word FOIL denotes F  First, O  Outer, I  Inner, L  Last. This method of multiplying binomials is restricted to binomials only and hence not applicable to all polynomial multiplications. The general form of the FOIL formula is: (a + b)(c + d) = ac + ad + bc + bd
Let us take two binomials (x + 2) and (x + 4) to understand the FOIL method of multiplying binomials. We will follow the sequence, First Outer Inner Last.
 Step 1: The first terms of both the binomials are taken and multiplied, i.e., x × x = x^{2}
 Step 2: Now, the first term of the first binomial and the second term of the second binomial also known as the outer terms of both the binomials will be multiplied, i.e., x × 4 = 4x
 Step 3: We will now consider the second term of the first binomial and the first term of the second binomial also known as the inner terms of both the binomials and multiply them, i.e., 2 × x = 2x
 Step 4: Finally, we will consider the second term of both the binomials also known as the last terms, and multiply them, i.e., 2 × 4 = 8
 Step 5: All the above results will be finally added and the like terms will be combined, i.e., x^{2} + 4x + 2x + 8 = x^{2} + 6x + 8
Thus, the product (x + 2)(x + 4) = x^{2} + 6x + 8
Multiplying Binomials Using the Vertical Method
Multiplying binomial using the vertical method is quite similar to the vertical multiplication of whole numbers. This method applies to all polynomial multiplications. Let's consider the binomials (x + 2) and (x + 3) and multiply them using the vertical method.
 Step 1: Place the binomials one below the other as shown in the figure.
 Step 2: Start with the second or the righthand term of the bottom binomial, i.e., 2, and multiply this value with both the terms of the top binomial individually that is (2 × x) + (2 × 3) = 2x + 6
 Step 3: Now let's consider the first or the lefthand term of the bottom binomial, i.e., x, and multiply this value with both the terms of the top binomial individually that is (x × x) + (3 × x) = x^{2} + 3x
 Step 4: Write the result obtained in the previous step in the second row in such a way that the like terms are lined up.
 Step 5: Finally, add the columns to obtain the result, i.e, x^{2} + 5x + 6
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Multiplying Binomial Examples

Example 1: Multiply the binomials (3x  5) and (4x + 2) using the distributive property.
Solution:
Let's use the distributive property of multiplying binomials for the given question.
(3x  5)(4x + 2)
= 3x(4x + 2)  5(4x + 2)
= (3x)(4x) + (3x)(2)  (5)(4x)  (5)(2)
= 12x^{2} + 6x  20x  10
= 12x^{2}  14x  10
Thus, (3x  5)(4x + 2) = 12x^{2}  14x  10

Example 2: Multiply the binomials (3a + 2b) and (5a + 7b) using the FOIL method.
Solution:
Let's use the FOIL method to multiply binomials for the given question. So, in (3a + 2b) (5a + 7b), we will follow the order: First, outer, inner, last.
When we multiply the first terms, we get 3a × 5a = 15a^{2}. After this, we multiply the outer terms, 3a × 7b = 21ab. Then, we multiply the inner terms, 2b × 5a = 10ab. Finally, we will multiply the last terms, 2b × 7b = 14b^{2}. Now, combining all the products together, 15a^{2 }+ 21ab + 10ab + 14b^{2}. After combining the like terms, we get 15a^{2} + 31ab + 14b^{2}. Hence, the product of (3a + 2b)(5a + 7b) is 15a^{2} + 31ab + 14b^{2}
FAQs on Multiplying Binomial
What is Multiplying Binomials?
An algebraic expression with two terms connected with a plus or a minus sign is known as a binomial. When such expressions are multiplied, this is termed as multiplying binomials. For example: (3x + 2)(5x + 1) means we need to multiply (3x + 2) and (5x + 1)
What does FOIL stand for in Multiplying Binomials?
For multiplying binomials, we use a method known as the FOIL method. The full form of FOIL is F  First, O  Outer, I  Inner, L  Last. So, FOIL(First Outer Inner Last) is the sequence in which the terms are multiplied in this order. In other words, the first terms are multiplied first, followed by the outer terms, then the inner terms, and then the last terms.
How to Multiply Binomials using Distributive Property?
Binomials are multiplied by applying the distributive property twice. Both the terms of the first binomial are individually multiplied with the second binomial, followed by the application of the distributive property to expand the terms. For example: Multiply (y + 7)(y + 3)
= y(y + 3) + 7(y + 3)
= y^{2} + 3y + 7y + 21
= y^{2} + 10y + 21
How to Multiply Binomials Without FOIL?
Apart from the method of FOIL(First, Outer, Inner, Last), binomials can be multiplied using various methods such as the distributive property and the vertical method of multiplying binomials.
How to Multiply 3 Binomials?
When 3 binomials are given, we multiply the first two binomials, calculate the product and multiply the third binomial with the product obtained.
For example: Multiply (x + 1)(x + 2)(x + 3). Here, let us multiply the first two binomials, x + 1 and x + 2 and then multiply the product by the third binomial.
= [x(x + 2) + 1(x + 2)] (x + 3)
= (x^{2} + 3x + 2)(x + 3)
= x(x^{2} + 3x + 2) + 3(x^{2} + 3x + 2)
= x^{3} + 3x^{2} + 2x + 3x^{2} + 9x + 6
= x^{3} + 6x^{2} + 11x + 6
How to Multiply Binomials and Trinomials?
Trinomials are those expressions that have 3 terms in its simplified form. Binomials can be multiplied by trinomials using the distributive property of multiplication.
For example: Multiply (x^{2} + 2x + 2)(x + 5)
= x(x^{2} + 2x + 2) + 5(x^{2} + 2x + 2)
= x^{3} + 2x^{2} + 2x + 5x^{2} + 10x + 10
= x^{3} + 7x^{2} + 12x + 10
How to Multiply a Binomial by a Monomial?
Monomials are those expressions that have a single term. Binomials can be multiplied by monomials using the distributive property of multiplication.
For example: Multiply 6x(2x + 1)
= (6x)(2x) + (6x)(1)
= 12x^{2} + 6x
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