Lesson Notes By Weeks and Term v3 - Junior Secondary 3
Factorization
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Factorization
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Subject: General Mathematics
Class: Junior Secondary 3
Term: 2nd Term
Week: 1
Theme: Algebraic Processes
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Factorize simple algebraic expressions Factorize quadratic algebraic expressions using quadratic equation box Solve word problems in volving factorization
"box"). Place $ax^2$ in the top-left cell. Place $c$ in the bottom-right cell. Place $mx$ and $nx$ in the remaining two cells (top-right and bottom-left, order doesn't matter). Find the HCF of each row and each column. The factors along the top and left sides of the box will be the two binomial factors.
Example 5: Factorize $x^2 + 7x + 10$. (Here $a=1, b=7, c=10$)
Step 1: $P = a \times c = 1 \times 10 = 10$. $S = b = 7$.
Step 2: Find two numbers that multiply to 10 and add to
7. These numbers are 2 and 5 ($2 \times 5 = 10$, $2 + 5 = 7$).
Step 3: Rewrite the middle term: $x^2 + 2x + 5x + 10$.
Step 4: Use the Box Method | | x | +5 | |---------|-----|-----| | x | $x^2$ | $5x$ | | +2 | $2x$ | $10$ | Row HCFs: HCF of $(x^2, 5x)$ is $x$. HCF of $(2x, 10)$ is $2$.
Column HCFs: HCF of $(x^2, 2x)$ is $x$. HCF of $(5x, 10)$ is $5$. The factors are $(x + 2)$ and $(x + 5)$.
Result: $(x + 2)(x + 5)$.
Example 6: Factorize $2y^2 - 11y + 12$. (Here $a=2, b=-11, c=12$)
Step 1: $P = a \times c = 2 \times 12 = 24$. $S = b = -11$.
Step 2: Find two numbers that multiply to 24 and add to -
1
1. These numbers are -3 and -8 ($-3 \times -8 = 24$, $-3 + -8 = -11$).
Step 3: Rewrite the middle term: $2y^2 - 3y - 8y + 12$.
Step 4: Use the Box Method | | 2y | -3 | |---------|------|-----| | y | $2y^2$ | $-3y$ | | -4 | $-8y$ | $12$ | Row HCFs: HCF of $(2y^2, -3y)$ is $y$. HCF of $(-8y, 12)$ is $-4$.
Column HCFs: HCF of $(2y^2, -8y)$ is $2y$. HCF of $(-3y, 12)$ is $-3$. The factors are $(y - 4)$ and $(2y - 3)$.
Result: $(y - 4)(2y - 3)$. C. Solving Word Problems involving Factorization This involves translating a given real-world problem into an algebraic expression (often a quadratic) and then factorizing it to find a solution.
Example 7 (Nigerian Context): A farmer in Kano wants to fence a rectangular plot of land. The area of the plot is given by the expression $x^2 + 10x + 24$ square metres. If the length of the plot is $(x+6)$ metres, find the expression for the width.
Understanding the Problem: Area of a rectangle = length $\times$ width. We are given the area and the length. We need to find the width.
Formulate: $x^2 + 10x + 24 = (x+6) \times \text{width}$. To find the width, we need to factorize the area expression $x^2 + 10x + 24$. Factorize $x^2 + 10x + 24$: $P = 1 \times 24 = 24$. $S = 10$. Two numbers that multiply to 24 and add to 10 are 4 and
6. Rewrite: $x^2 + 4x + 6x + 24$.
Using the Box Method: | | x | +6 | |---------|-----|-----| | x | $x^2$ | $6x$ | | +4 | $4x$ | $24$ | Factors are $(x+4)(x+6)$.
Solve: So, the area is $(x+4)(x+6)$. Since Area = Length $\times$ Width, and we have $(x+6)$ as length, the width must be $(x+4)$. * Answer: The expression for the width is $(x+4)$ metres. --- Factorization is the reverse process of expansion. It involves finding two or more expressions whose product is the original expression.
This lesson focuses on two main types: factoring simple algebraic expressions (involving common factors or grouping) and factoring quadratic expressions. A. Factorization of Simple Algebraic Expressions This involves identifying common factors among the terms of an expression and then rewriting the expression as a product of the common factor and the remaining terms.
1. Finding the Highest Common Factor (HCF): Identify the HCF of the numerical coefficients. Identify the HCF of the variables (the lowest power of each common variable). Multiply these HCFs to get the HCF of the terms.
Example 1: Factorize $6x + 9y$. HCF of 6 and 9 is
3. No common variable.
Therefore, the HCF of $6x$ and $9y$ is
3. Divide each term by the HCF: $6x \div 3 = 2x$, $9y \div 3 = 3y$. Write the expression as the product of the HCF and the remaining terms: $3(2x + 3y)$.
Example 2: Factorize $12a^2b - 18ab^3$. HCF of 12 and 18 is
6. HCF of $a^2$ and $a$ is $a$. HCF of $b$ and $b^3$ is $b$.
Therefore, the HCF of $12a^2b$ and $18ab^3$ is $6ab$. Divide each term by $6ab$: $12a^2b \div 6ab = 2a$ $-18ab^3 \div 6ab = -3b^2$ Result: $6ab(2a - 3b^2)$.
2. Factorization by Grouping: This method is used when an expression has four terms, and a common factor can be found in pairs of terms.
Example 3: Factorize $ax + ay + bx + by$.
Group the terms into two pairs: $(ax + ay) + (bx + by)$. Factor out the common factor from each pair: From $(ax + ay)$, the common factor is $a$, leaving $a(x + y)$. From $(bx + by)$, the common factor is $b$, leaving $b(x + y)$. Now the expression is $a(x + y) + b(x + y)$. Notice that $(x + y)$ is a common factor to both terms. Factor out $(x + y)$: $(x + y)(a + b)$.
Example 4: Factorize $6pq - 9q + 2pr - 3r$.
Group: $(6pq - 9q) + (2pr - 3r)$.
Factor each pair: $3q(2p - 3)$ $r(2p - 3)$ Expression becomes $3q(2p - 3) + r(2p - 3)$. Factor out the common bracket $(2p - 3)$: $(2p - 3)(3q + r)$. B. Factorization of Quadratic Algebraic Expressions using the Quadratic Equation Box A quadratic expression is a polynomial of degree 2, generally in the form $ax^2 + bx + c$, where $a, b, c$ are constants and $a \neq 0$. The "quadratic equation box" method (also known as the diamond-box method or product-sum method combined with a grid) is a systematic way to factorize these expressions. Steps for Factorizing $ax^2 + bx + c$ using the Quadratic Equation Box:
1. Find the product (P) and sum (S): Product (P) = $a \times c$ (coefficient of $x^2$ multiplied by the constant term). Sum (S) = $b$ (coefficient of the $x$ term).
2. Find two numbers: Identify two numbers (let's call them $m$ and $n$) that multiply to P and add up to S. $m \times n = P$ $m + n = S$
3. Rewrite the middle term: Replace the middle term $bx$ with $mx + nx$. The expression becomes $ax^2 + mx + nx + c$.
4. Factor by Grouping (using the box): Draw a 2x2 grid (the "box"). Place $ax^2$ in the top-left cell. Place $c$ in the bottom-right cell. Place $mx$ and $nx$ in the remaining two cells (top-right and bottom-left, order doesn't matter). Find the HCF of each row and each column. The factors along the top and left sides of the box will be the two binomial factors.
Example 5: Factorize $x^2 + 7x + 10$. (Here $a=1, b=7, c=10$)
Step 1: $P = a \times c = 1 \times 10 = 10$. $S = b = 7$. *Step Introduction (10 minutes)
Teacher Activity: Begin by reviewing the concept of expansion (e.g., $(x+2)(x+3) = x^2+5x+6$). Explain that factorization is the reverse process. Ask students to identify common factors in simple number sequences (e.g., 6, 9, 12 – common factor 3). Introduce the term "factorization" as breaking down an expression into its multiplicative components.
Student Activity: Students engage in a brief recall of algebraic expansion. They identify common factors in numerical examples.
Activity 1: Factorizing Simple Algebraic Expressions (20 minutes)
Teacher Activity: Demonstrate finding the HCF for terms with both numbers and variables (e.g., $10x - 15y$, $8ab + 12a^2$). Guide students through identifying the numerical HCF and variable HC
F. Provide several examples on the board, ensuring steps are clear: identify HCF, divide each term by HCF, write in factored form. Introduce factorization by grouping with a four-term expression (e.g., $ac + ad + bc + bd$). Emphasize identifying common brackets after the first round of factorization. Circulate to monitor understanding and provide individual support.
Student Activity: Work with a partner to factorize expressions with common factors. Practice factorization by grouping with provided examples. Present their solutions on the board or to the class, explaining their steps.
Activity 2: Factorizing Quadratic Algebraic Expressions using the Quadratic Equation Box (30 minutes)
Teacher Activity: Clearly introduce the standard form of a quadratic expression ($ax^2 + bx + c$). Explain the "Product-Sum" method: finding two numbers that multiply to $ac$ and add to $b$. Emphasize the importance of signs. Demonstrate the quadratic equation box method step-by-step using an example where $a=1$ (e.g., $x^2 + 6x + 8$). Fill in the box visibly and explain how row and column HCFs lead to the factors. Demonstrate a more complex example where $a>1$ (e.g., $3x^2 - 10x + 8$). Stress careful handling of negative signs. Allow students to try a similar problem individually or in pairs. Address common misconceptions, such as incorrect sign combinations or incomplete factorization.
Student Activity: Practice identifying $a, b, c$ for given quadratic expressions. Work collaboratively to find the "product" and "sum" numbers for various quadratics. Individually attempt to factorize quadratic expressions using the quadratic box method on their exercise books or mini whiteboards. Compare answers and discuss discrepancies within their groups.
Activity 3: Solving Word Problems involving Factorization (20 minutes)
Teacher Activity: Present a word problem on the board (e.g., involving area or dimensions of a farm plot). Guide students through translating the word problem into an algebraic expression. Demonstrate how to factorize the derived expression using previously learned methods. Show how the factors relate back to the context of the problem (e.g., length and width). Provide another word problem for students to solve in small groups, encouraging them to discuss translation, factorization, and interpretation.
Student Activity: Read and interpret word problems. Work in groups to translate word problems into algebraic expressions. Apply factorization techniques to solve the algebraic expressions. Relate the factored results back to the original problem context and present their findings. Conclusion & Wrap-up (5 minutes)
Teacher Activity: Briefly recap the different methods of factorization covered (common factor, grouping, quadratic box). Emphasize their importance in simplifying expressions and solving problems. Assign homework.
Student Activity: Participate in a quick Q&A session, summarizing key takeaways. Note down homework. --- Question 1 (Targeting Objective 1: Simple algebraic expressions)
Factorize completely: $15xy - 20yz$ Context: A small business in Aba produces textiles. They incurred costs for dyeing ($15xy$) and weaving ($20yz$). Find the common factor in their costs.
Solution 1:
1. Identify HCF of numerical coefficients: HCF of 15 and 20 is 5.
2. Identify HCF of variables: Both terms have 'y'. The lowest power of y is $y^1$. 'x' and 'z' are not common to both. So, HCF of variables is $y$.
3. Overall HCF: $5y$.
4. Divide each term by HCF: $15xy \div 5y = 3x$ $-20yz \div 5y = -4z$
5. Write in factored form: $5y(3x - 4z)$
Commentary: This solution demonstrates the identification of both numerical and variable HCFs, a fundamental step in simple factorization. The context helps relate the algebraic expression to a common Nigerian business scenario.
Question 2 (Targeting Objective 1: Factorization by grouping)
Factorize: $3m + 3n - pm - pn$ Context: A community development project aims to distribute 'm' bags of cement and 'n' bags of sand. If three volunteers (3m + 3n) work on it and 'p' people use 'm' and 'n' quantities (-pm - pn), how can the total effort be expressed in factored form?
Solution 2:
1. Group the terms: $(3m + 3n) - (pm + pn)$.
Note: Be careful with the minus sign when grouping.
2. Factor out common factors from each group: From $(3m + 3n)$, common factor is 3: $3(m + n)$. From $-(pm + pn)$, common factor is $-p$: $-p(m + n)$.
3. The expression becomes: $3(m + n) - p(m + n)$.
4. Factor out the common binomial factor: $(m + n)(3 - p)$.
Commentary: This problem emphasizes careful handling of negative signs during grouping. The result shows that the overall effort can be simplified into two components.
Question 3 (Targeting Objective 2: Factorizing quadratic expressions, a=1) Factorize $p^2 - 13p + 30$ using the quadratic equation box method.
Context: A carpenter in Lagos is designing a rectangular table top. The area of the top is given by $p^2 - 13p + 30$ square inches. Find the expressions for its length and width.
Solution 3:
1. Identify P and S: $a=1, b=-13, c=30$. Product (P) = $a \times c = 1 \times 30 = 30$. Sum (S) = $b = -13$.
2. Find two numbers: Numbers that multiply to 30 and add to -13 are -3 and -10. $(-3) \times (-10) = 30$ $(-3) + (-10) = -13$
3. Rewrite the middle term: $p^2 - 3p - 10p + 30$.
4. Use the Box Method: | | p | -10 | |---------|-----|------| | p | $p^2$ | $-10p$ | | -3 | $-3p$ | $30$ | Row HCFs: HCF of $(p^2, -10p)$ is $p$. HCF of $(-3p, 30)$ is $-3$.
Column HCFs: HCF of $(p^2, -3p)$ is $p$. HCF of $(-10p, 30)$ is $-10$.
5. Factors: $(p - 3)$ and $(p - 10)$.
Commentary: This solution walks through the quadratic box method clearly, including identifying P and S, finding the correct numbers, and filling the box. It directly applies to finding dimensions from an area expression.
Question 4 (Targeting Objective 2: Factorizing quadratic expressions, a>1) Factorize $3x^2 + 14x + 8$ using the quadratic equation box method.
Context: An architect is calculating the surface area of a specialized solar panel, given by $3x^2 + 14x + 8$ square units. Determine the expressions for its dimensions.
Solution 4:
1. Identify P and S: $a=3, b=14, c=8$. Product (P) = $a \times c = 3 \times 8 = 24$. Sum (S) = $b = 14$.
2. Find two numbers: Numbers that multiply to 24 and add to 14 are 2 and 12. $2 \times 12 = 24$ $2 + 12 = 14$
3. Rewrite the middle term: $3x^2 + 2x + 12x + 8$.
4. Use the Box Method: | | 3x | +2 | |---------|------|-----| | x | $3x^2$ | $2x$ | | +4 | $12x$ | $8$ | Row HCFs: HCF of $(3x^2, 2x)$ is
Land Use and Urban Planning (e.g., Abuja Master Plan): Factorization is implicitly used when designing layouts for housing estates, markets, or public spaces. If the area of a proposed rectangular park is represented by a quadratic expression, factoring it helps urban planners determine the possible length and width dimensions. For instance, if a park has an area of $(x^2 + 13x + 40)$ sq. units, factorization to $(x+5)(x+8)$ gives potential dimensions of $(x+5)$ and $(x+8)$ units. This informs decisions on land demarcation, infrastructure placement, and resource allocation within a given area. Budgeting and Resource Management (Household/Small Business): A Nigerian shop owner might need to optimize packaging for products like garri or rice. If the total cost of packaging for two items is expressed as $10x + 15y$ (where $x$ and $y$ are quantities and ₦10 and ₦15 are unit costs), factoring it to $5(2x + 3y)$ helps to identify the common factor (₦5, perhaps representing a base cost or material unit) that links the two costs. This aids in understanding cost structures and finding ways to reduce expenses by identifying common denominators in different expenditures. Engineering and Construction (Bridge/Building Design): When calculating the stress distribution on structural components (like beams or columns), engineers often encounter quadratic equations. For example, the load-bearing capacity of a beam might be described by an expression like $2w^2 - 5w + 3$. Factorizing this expression to $(2w-3)(w-1)$ could help determine critical points or values of 'w' (e.g., width or weight parameter) that affect the beam's stability. This is crucial for ensuring the safety and structural integrity of infrastructure projects like the 3rd Mainland Bridge or the Eko Bridge. ---