Factorize The Following Expression:$(w - 12)(w - 5) = 0$Find The Values Of $w$:$w = 12$ Or $w = 5$

Introduction

In algebra, factorizing expressions is a crucial concept that helps us simplify complex equations and solve for unknown variables. In this article, we will focus on factorizing the expression $(w - 12)(w - 5) = 0$ and find the values of $w$ that satisfy this equation.

What is Factorizing?

Factorizing is the process of expressing an algebraic expression as a product of simpler expressions, called factors. These factors can be numbers, variables, or a combination of both. The goal of factorizing is to break down a complex expression into smaller, more manageable parts that can be easily solved or manipulated.

The Expression to be Factorized

The given expression is $(w - 12)(w - 5) = 0$. This expression consists of two binomial factors, $(w - 12)$ and $(w - 5)$, which are multiplied together to form a quadratic expression.

Step 1: Apply the Zero Product Property

The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. In this case, we have:

$$(w - 12)(w - 5) = 0$$

Using the zero product property, we can set each factor equal to zero and solve for $w$:

$$(w - 12) = 0 \quad \text{or} \quad (w - 5) = 0$$

Step 2: Solve for $w$

Now, we need to solve each equation for $w$:

$$(w - 12) = 0 \quad \Rightarrow \quad w = 12$$

$$(w - 5) = 0 \quad \Rightarrow \quad w = 5$$

Therefore, the values of $w$ that satisfy the given equation are $w = 12$ or $w = 5$.

Conclusion

In this article, we factorized the expression $(w - 12)(w - 5) = 0$ and found the values of $w$ that satisfy this equation. We applied the zero product property to break down the expression into simpler factors and then solved each equation for $w$. The final answer is $w = 12$ or $w = 5$.

Real-World Applications

Factorizing algebraic expressions has numerous real-world applications in various fields, including:

• Physics: Factorizing expressions is used to solve equations of motion, energy, and momentum.
• Engineering: Factorizing expressions is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: Factorizing expressions is used in algorithms and data structures, such as graph theory and combinatorics.

Tips and Tricks

Here are some tips and tricks to help you factorize algebraic expressions:

• Look for common factors: Identify common factors in the expression and factor them out.
• Use the zero product property: Apply the zero product property to break down the expression into simpler factors.
• Use algebraic identities: Use algebraic identities, such as the difference of squares, to factorize expressions.

Common Algebraic Identities

Here are some common algebraic identities that can be used to factorize expressions:

• Difference of squares: $a^2 - b^2 = (a - b)(a + b)$
• Sum of squares: $a^2 + b^2 = (a + b)^2 - 2ab$
• Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Conclusion

Q&A: Factorizing Algebraic Expressions

Q: What is factorizing in algebra?

A: Factorizing is the process of expressing an algebraic expression as a product of simpler expressions, called factors. These factors can be numbers, variables, or a combination of both.

Q: Why is factorizing important in algebra?

A: Factorizing is important in algebra because it helps us simplify complex equations and solve for unknown variables. By breaking down an expression into simpler factors, we can easily identify the values of variables that satisfy the equation.

Q: What are some common algebraic identities used in factorizing?

A: Some common algebraic identities used in factorizing include:

• Difference of squares: $a^2 - b^2 = (a - b)(a + b)$
• Sum of squares: $a^2 + b^2 = (a + b)^2 - 2ab$
• Difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Q: How do I apply the zero product property in factorizing?

A: To apply the zero product property, you need to set each factor equal to zero and solve for the variable. For example, if you have the expression $(w - 12)(w - 5) = 0$, you can set each factor equal to zero and solve for $w$:

$$(w - 12) = 0 \quad \Rightarrow \quad w = 12$$

$$(w - 5) = 0 \quad \Rightarrow \quad w = 5$$

Q: What are some tips and tricks for factorizing algebraic expressions?

A: Here are some tips and tricks for factorizing algebraic expressions:

• Look for common factors: Identify common factors in the expression and factor them out.
• Use the zero product property: Apply the zero product property to break down the expression into simpler factors.
• Use algebraic identities: Use algebraic identities, such as the difference of squares, to factorize expressions.

Q: Can you give an example of factorizing a quadratic expression?

A: Yes, here's an example of factorizing a quadratic expression:

Suppose we have the quadratic expression $x^2 + 5x + 6$. We can factorize this expression as follows:

$$x^2 + 5x + 6 = (x + 2)(x + 3)$$

Q: How do I factorize a quadratic expression with a negative sign?

A: To factorize a quadratic expression with a negative sign, you need to use the zero product property and set each factor equal to zero. For example, if you have the expression $-(x + 2)(x + 3) = 0$, you can set each factor equal to zero and solve for $x$:

$$(x + 2) = 0 \quad \Rightarrow \quad x = -2$$

$$(x + 3) = 0 \quad \Rightarrow \quad x = -3$$

Q: Can you give an example of factorizing a polynomial expression?

A: Yes, here's an example of factorizing a polynomial expression:

Suppose we have the polynomial expression $x^3 - 2x^2 - 5x + 6$. We can factorize this expression as follows:

$$x^3 - 2x^2 - 5x + 6 = (x - 1)(x^2 - x - 6)$$

Conclusion

In conclusion, factorizing algebraic expressions is a crucial concept in algebra that helps us simplify complex equations and solve for unknown variables. By applying the zero product property and using algebraic identities, we can factorize expressions and find the values of variables that satisfy the equation.