Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Algebra → Identities and Simplification ↓


Multiplying Binomials


In the world of algebra, a key concept is multiplying binomials. A binomial is an expression that consists of two terms. For example, (x + 2) and (y - 3) are examples of binomials. When you multiply two binomials, you use the distributive property to ensure that each term in the first binomial is multiplied by each term in the second binomial. This concept is not only fundamental in algebra, but is also a cornerstone in simplifying expressions and solving equations.

Understanding binomials

Let us first understand what a binomial is. As mentioned earlier, a binomial is a polynomial with only two terms. Examples include:

  • (x + 3)
  • (a - b)
  • (2y + 5)

How to multiply binomials

When we multiply binomials, we use a method often called "FOIL." It stands for first, outer, inner, and last. Each of these terms refers to the pair of terms you multiply together.

Foil method

Consider multiplying the binomials (x + 2) and (x + 3). Using the FOIL method, we would do:

  1. First: Multiply the first terms of each binomial: x * x = x 2.
  2. External: Multiply the external terms: x * 3 = 3x.
  3. Inner: Multiply the inner terms: 2 * x = 2x.
  4. Final: Multiply the last terms: 2 * 3 = 6.

Now, add all of these products together: x 2 + 3x + 2x + 6 Combine like terms (in this case, 3x and 2x):

x 2 + 3x + 2x + 6 = x 2 + 5x + 6

Visualizing binomial multiplication

Sometimes, visualizing the multiplication can help you understand the process better. Here is a representation of the expression we multiplied:

  (x + 2)
 x   3
 ,    
     → forms a product with the combinations x + 2 and x + 3:
      (x times x) + (x times 3) + (2 times x) + (2 times 3)

Write it clearly:

  | x 3
, 
x | x 2  3x
2 | 2x 6

This helps to make sure that you understand how each part of (x + 2) is interacting with each part of (x + 3) during the multiplication.

Example with detailed steps

Example 1: Multiply (x + 4)(x - 5)

Let's multiply these two binomials using the FOIL method.

1. First: x * x = x 2

2. External: x * (-5) = -5x

3. Inner: 4 * x = 4x

4. Final: 4 * (-5) = -20

Now, add these results together:

x 2 - 5x + 4x - 20

Combine like terms:

x 2 - x - 20

Example 2: Multiply (2m + 3)(3m - 1)

Let us analyze this.

1. First: 2m * 3m = 6m 2

2. External: 2m * (-1) = -2m

3. Interior: 3 * 3m = 9m

4. Final: 3 * (-1) = -3

Add these:

6m 2 - 2m + 9m - 3

Combine like terms:

6m 2 + 7m - 3

Practice problems

  1. Multiply (y + 2)(y - 6).
  2. Multiply (3p - 1)(2p + 5).
  3. Multiply (a - 4)(a + 3).
  4. Multiply (x - 2)(x + 7).

Why does it matter?

Multiplying binomials is an important skill in algebra because it forms the basis for understanding the multiplication of polynomials in more complex situations. This operation is often used to expand expressions, simplify them, and solve quadratic equations. Knowing how to perform it fluently helps students approach more advanced algebraic concepts with confidence.

Additional tips

Here are some additional tips to keep in mind when multiplying binomials:

  • Always put the correct sign on all the terms during multiplication.
  • Combine like terms carefully to avoid mistakes in the final expression.
  • Practice regularly to become proficient at the FOIL method and identifying words.

Remember, mastering multiplying binomials makes the journey through algebra even easier. With consistent practice, this method will become second nature, paving the way for tackling more advanced math with confidence.


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Multiplying Binomials

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