Simplify The Polynomial:$ (w+6)(3w-4) $
Introduction
In algebra, simplifying polynomials is an essential skill that helps in solving equations and manipulating expressions. A polynomial is an expression consisting of variables and coefficients combined using only addition, subtraction, and multiplication. In this article, we will focus on simplifying the given polynomial expression: (w+6)(3w-4). We will use the distributive property to expand and simplify the expression.
Understanding the Distributive Property
The distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions. It states that for any real numbers a, b, and c:
a(b + c) = ab + ac
This property can be applied to any expression, including polynomials. We will use this property to simplify the given polynomial expression.
Expanding the Polynomial Expression
To simplify the polynomial expression (w+6)(3w-4), we will use the distributive property. We will multiply each term in the first expression (w+6) by each term in the second expression (3w-4).
(w+6)(3w-4) = w(3w-4) + 6(3w-4)
Applying the Distributive Property
Now, we will apply the distributive property to each term in the expression.
w(3w-4) = 3w^2 - 4w 6(3w-4) = 18w - 24
Combining Like Terms
Now that we have expanded the expression, we can combine like terms. Like terms are terms that have the same variable raised to the same power.
3w^2 - 4w + 18w - 24
Simplifying the Expression
Now, we can simplify the expression by combining like terms.
3w^2 - 4w + 18w - 24 = 3w^2 + 14w - 24
Conclusion
In this article, we simplified the polynomial expression (w+6)(3w-4) using the distributive property. We expanded the expression, applied the distributive property, and combined like terms to simplify the expression. The final simplified expression is 3w^2 + 14w - 24.
Final Answer
The final answer is 3w^2 + 14w - 24.
Example Use Case
Simplifying polynomials is an essential skill in algebra that helps in solving equations and manipulating expressions. Here's an example use case:
Suppose we have the equation:
(w+6)(3w-4) = 0
To solve this equation, we can simplify the left-hand side of the equation using the distributive property.
(w+6)(3w-4) = 3w^2 + 14w - 24
Now, we can set the simplified expression equal to zero and solve for w.
3w^2 + 14w - 24 = 0
This is a quadratic equation that can be solved using various methods, including factoring, the quadratic formula, or graphing.
Tips and Tricks
Here are some tips and tricks for simplifying polynomials:
- Use the distributive property to expand and simplify expressions.
- Combine like terms to simplify the expression.
- Use the quadratic formula to solve quadratic equations.
- Factor expressions to simplify them.
Common Mistakes
Here are some common mistakes to avoid when simplifying polynomials:
- Failing to apply the distributive property.
- Failing to combine like terms.
- Making errors when applying the distributive property.
- Failing to check the final answer.
Conclusion
Simplifying polynomials is an essential skill in algebra that helps in solving equations and manipulating expressions. In this article, we simplified the polynomial expression (w+6)(3w-4) using the distributive property. We expanded the expression, applied the distributive property, and combined like terms to simplify the expression. The final simplified expression is 3w^2 + 14w - 24. We also provided an example use case and tips and tricks for simplifying polynomials.
Introduction
In our previous article, we simplified the polynomial expression (w+6)(3w-4) using the distributive property. We expanded the expression, applied the distributive property, and combined like terms to simplify the expression. In this article, we will answer some frequently asked questions (FAQs) related to simplifying polynomials.
Q&A
Q: What is the distributive property?
A: The distributive property is a fundamental concept in algebra that allows us to expand and simplify expressions. It states that for any real numbers a, b, and c:
a(b + c) = ab + ac
This property can be applied to any expression, including polynomials.
Q: How do I apply the distributive property to simplify a polynomial expression?
A: To apply the distributive property, you need to multiply each term in the first expression by each term in the second expression. For example, to simplify the expression (w+6)(3w-4), you would multiply w by 3w and w by -4, and then multiply 6 by 3w and 6 by -4.
Q: What are like terms?
A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x are like terms because they both have the variable x raised to the power of 1.
Q: How do I combine like terms?
A: To combine like terms, you need to add or subtract the coefficients of the like terms. For example, to combine the terms 2x and 4x, you would add the coefficients 2 and 4 to get 6x.
Q: What is the final simplified expression for (w+6)(3w-4)?
A: The final simplified expression for (w+6)(3w-4) is 3w^2 + 14w - 24.
Q: Can I use the distributive property to simplify any polynomial expression?
A: Yes, you can use the distributive property to simplify any polynomial expression. However, you need to make sure that you are applying the property correctly and combining like terms.
Q: What are some common mistakes to avoid when simplifying polynomials?
A: Some common mistakes to avoid when simplifying polynomials include:
- Failing to apply the distributive property
- Failing to combine like terms
- Making errors when applying the distributive property
- Failing to check the final answer
Q: How do I check the final answer?
A: To check the final answer, you need to plug the simplified expression back into the original equation and make sure that it is true. For example, if you simplified the expression (w+6)(3w-4) to 3w^2 + 14w - 24, you would plug this expression back into the original equation and make sure that it is true.
Example Use Case
Suppose we have the equation:
(w+6)(3w-4) = 0
To solve this equation, we can simplify the left-hand side of the equation using the distributive property.
(w+6)(3w-4) = 3w^2 + 14w - 24
Now, we can set the simplified expression equal to zero and solve for w.
3w^2 + 14w - 24 = 0
This is a quadratic equation that can be solved using various methods, including factoring, the quadratic formula, or graphing.
Tips and Tricks
Here are some tips and tricks for simplifying polynomials:
- Use the distributive property to expand and simplify expressions.
- Combine like terms to simplify the expression.
- Use the quadratic formula to solve quadratic equations.
- Factor expressions to simplify them.
Conclusion
Simplifying polynomials is an essential skill in algebra that helps in solving equations and manipulating expressions. In this article, we answered some frequently asked questions (FAQs) related to simplifying polynomials. We also provided an example use case and tips and tricks for simplifying polynomials.