Multiplying Expressions (Junior Cert Mathematics): Revision Notes

Worked Example 1: Multiplying Simple Terms Problem: Multiply $3x$ by $4x$.

Step 1: Multiply the coefficients

• The coefficients are $3$ and $4$. Multiply them: $3 \times 4 = :success[12]$

Step 2: Multiply the variables

• Both terms have the variable $x$. When multiplying $x \times x$, you add the exponents. Since each $x$ has an exponent of $1$ (which is usually not written), you add the exponents: $x^1 \times x^1 = x^{1+1} = :success[x^2]$

Step 3: Combine the results

• The final expression is: $:success[12x^2]$

Explanation: Here, we multiplied the numbers (coefficients) first, getting $12$. Then, we multiplied the variables, remembering to add their exponents. This gives us $x^2$, so the final answer is $12x^2$.

Worked Example 2: Multiplying a Number by a Binomial A binomial is an expression with two terms, like $x + 5$.

Problem: Multiply $2 \times (x + 5)$.

Step 1: Distribute the number to each term in the binomial

• Multiply the number 2 by each term inside the bracket: $[ 2 \times x = :success[2x] ] [ 2 \times 5 = :success[10] ]$

Step 2: Combine the results

• The final expression is: $:success[2x + 10]$

Explanation: When multiplying a number by a binomial, you apply the distributive property, which means multiplying the number outside the bracket by each term inside the bracket separately. This step-by-step approach ensures that you correctly multiply every part of the expression.

Worked Example 3: Multiplying Two Binomials Using the "Split and Repeat" Method This method is an efficient way to handle multiplying two binomials, like $(x + 3)(x - 2)$.

Problem: Multiply $(x + 3)(x - 2) .$

Step 1: Split the brackets

• $(x+3)(x-2)$
• $x(x-2)+3(x-2)$ Take the first term from the first binomial,$x ,$and multiply it by the entire second binomial $x - 2 .$

Then, take the second term from the first binomial,$+3$, and multiply it by the entire second binomial $(x - 2) .$

Step 2: Multiply each term

• Multiply $x$ by each term in the second binomial: $[ x \times x = :success[x^2] ] [ x \times -2 = :success[-2x] ]$
• Multiply $3$ by each term in the second binomial: $[ 3 \times x = :success[3x] ] [ 3 \times -2 = :success[-6] ]$

Step 3: Combine like terms

• Now, add the results together: $x^2 - 2x + 3x - 6$
• Combine the like terms $( -2x )$ and $( 3x )$: $:success[x^2 + x - 6]$

Final Answer: The expression simplifies to: $:success[x^2 + x - 6]$

Explanation: The "split and repeat" method is a systematic way to ensure you multiply every term in the first binomial by every term in the second. After you multiply, the final step is to combine any like terms, which in this case were $-2x$ and $3x .$

Example 4: Multiplying Two Binomials This example is more advanced because you'll need to multiply two expressions that each have two terms.

Problem: Multiply $(x + 2)$ by $(x + 3)$.

Step 1: Use the distributive property (FOIL method)

• Multiply the First terms: $x \times x = :success[x^2]$.
• Multiply the Outer terms: $x \times 3 = :success[3x]$.
• Multiply the Inner terms: $2 \times x = :success[2x]$.
• Multiply the Last terms:$2 \times 3 = :success[6] .$

Step 2: Combine like terms

• Combine $3x$ and $2x$: $x^2 + 3x + 2x + 6 = :success[x^2 + 5x + 6]$

Step 3: Write the result

• The final expression is: $:success[x^2 + 5x + 6]$

Explanation: This example shows how to multiply two binomials using the FOIL method (First, Outer, Inner, Last). After multiplying each pair of terms, you combine the like terms. This type of problem often appears in exams because it checks your understanding of both multiplication and combining like terms.

Example 5 Multiplying Binomials Using the "Split and Repeat" Method Problem: Multiply $(x + 3)(x - 2).$

Step 1: Split the brackets

• First, take the first term from the first binomial, $x ,$and multiply it by the entire second binomial: $x(x - 2)$
• Next, take the second term from the first binomial,$+3$, and multiply it by the entire second binomial: $+3(x - 2)$
• $x(x-2)+3(x-2)$

Step 2: Multiply each part

• Multiply$( x )$ by each term in the second binomial: $x(x-2)$ $x \times x = :success[x^2]$ $x \times -2 = :success[-2x]$
• Multiply $( +3 )$ by each term in the second binomial: $3(x-2)$ $3 \times x = :success[3x]$ $3 \times -2 = :success[-6]$

Step 3: Combine like terms

• Add all the terms together: $x^2 - 2x + 3x - 6$
• Combine $-2x$ and $3x$ to simplify: $:success[x^2 + x - 6]$

Final Answer: The expression simplifies to: $:success[x^2 + x - 6]$

Explanation: By splitting the multiplication into parts (i.e., multiplying each term from the first binomial by the entire second binomial), you can clearly see how each term contributes to the final expression. This method is particularly useful because it visually shows the examiner the process that you followed. Even if you do not reach the correct answer, the examiner will be able to give you attempt marks.

Problem 1: Multiply the following expressions

Question : Multiply $( 5x )$ by $( 3x ).$

Problem 2: Expand the expression

Question : Expand and simplify $4(y + 7)$

Problem 3: Multiply the binomials

Question : Multiply and simplify the expression $(x + 4)(x - 3) .$

Problem 4: Multiply the following and simplify

Question : Expand and simplify $3(2x - 5).$

Problem 5: Multiply and simplify the expression

Question : Multiply and simplify the expression $(2x + 1)(x - 6).$

Worked Example 1: Problem: Multiply $5x$ by $3x$.

Step 1: Multiply the coefficients

• Multiply the numbers (coefficients) first: $5 \times 3 = :success[15]$

Step 2: Multiply the variables

• Multiply the variables $( x \times x )$: $x \times x = :success[x^2]$

Step 3: Combine the results

• The final expression is: $:success[15x^2]$

Explanation: Here, you first multiplied the coefficients ($5$ and $3$) to get $15$. Then you multiplied the variables $( x \times x)$, which gives$( x^2 )$. So the final answer is $( 15x^2 ).$

Worked Example 2: Problem: Expand and simplify $4(y + 7)$.

Step 1: Distribute the number

• Start by multiplying the number outside the bracket by each term inside the bracket: $4(y + 7)$
• This splits into: $4 \times y + 4 \times 7$

Step 2: Multiply each part

• Multiply: $[ 4 \times y = :success[4y] ] [ 4 \times 7 = :success[28] ]$

Step 3: Combine the results

• The final expression is: $:success[4y + 28]$

Explanation: The distributive property was applied by multiplying $4$ by both $y$ and $7$. The steps were shown explicitly, ensuring clarity.

Worked Example 3: Problem: Multiply and simplify $(x + 4)(x - 3)$.

Step 1: Split the brackets

• Multiply the first term from the first binomial by the entire second binomial: $x(x - 3)$
• Multiply the second term from the first binomial by the entire second binomial: $+4(x - 3)$
• Now combine: $x(x - 3) + 4(x - 3)$

Step 2: Multiply each part

• Distribute $( x )$ through $(x - 3)$: $[ x \times x = :success[x^2] ] [ x \times -3 = :success[-3x] ]$
• Distribute $( 4 )$ through $(x - 3)$: $[ 4 \times x = :success[4x] ] [ 4 \times -3 = :success[-12] ]$

Step 3: Combine like terms

• Add all the terms together: $x^2 - 3x + 4x - 12$
• Combine $( -3x )$ and $( 4x )$: $:success[x^2 + x - 12]$

Final Answer: The expression simplifies to: $:success[x^2 + x - 12]$

Explanation: By splitting the binomials and carefully multiplying each term, every step is made clear. The combination of like terms at the end simplifies the expression.

Worked Example 4: Problem: Expand and simplify $3(2x - 5)$.

Step 1: Distribute the number

• Multiply the number outside the bracket by each term inside: $3(2x - 5)$
• This splits into: $3 \times 2x + 3 \times -5$

Step 2: Multiply each part

• Multiply: $[ 3 \times 2x = :success[6x] ] [ 3 \times -5 = :success[-15] ]$

Step 3: Combine the results

• The final expression is: $:success[6x - 15]$

Explanation: Each multiplication step was shown clearly, ensuring that every term was accounted for in the expansion.

Worked Example 5: Problem: Multiply and simplify $(2x + 1)(x - 6) .$

Step 1: Split the brackets

• Multiply the first term from the first binomial by the entire second binomial: $2x(x - 6)$
• Multiply the second term from the first binomial by the entire second binomial: $+1(x - 6)$
• Now combine: $2x(x - 6) + 1(x - 6)$

Step 2: Multiply each part

• Distribute $2x$ through $(x - 6)$: $[ 2x \times x = :success[2x^2] ] [ 2x \times -6 = :success[-12x] ]$
• Distribute $1$ through $(x - 6)$: $[ 1 \times x = :success[x] ] [ 1 \times -6 = :success[-6] ]$

Step 3: Combine like terms

• Add all the terms together: $2x^2 - 12x + x - 6$
• Combine $-12x$ and $( x )$: $:success[2x^2 - 11x - 6]$

Final Answer: The expression simplifies to: $:success[2x^2 - 11x - 6]$

Explanation: By splitting the binomials and multiplying each term carefully, you ensure that every part of the expression is clear. The final step of combining like terms leads to the simplified expression.