Simplify:\[$(x+4)(2+3x)\$\]
Simplify: {(x+4)(2+3x)$}$
In algebra, simplifying expressions is a crucial step in solving equations and inequalities. It involves combining like terms and removing any unnecessary components from the expression. In this article, we will simplify the given expression {(x+4)(2+3x)$}$, which is a product of two binomials.
Understanding the Expression
The given expression is a product of two binomials: ($(x+4)$ and $(2+3x)$. To simplify this expression, we need to multiply each term in the first binomial by each term in the second binomial.
Multiplying Binomials
When multiplying binomials, we use the distributive property, which states that for any real numbers a, b, c, and d:
a(b + c) = ab + ac
Using this property, we can multiply each term in the first binomial by each term in the second binomial.
Step 1: Multiply the First Term in the First Binomial by Each Term in the Second Binomial
The first term in the first binomial is x. We multiply x by each term in the second binomial:
x(2) = 2x x(3x) = 3x^2
Step 2: Multiply the Second Term in the First Binomial by Each Term in the Second Binomial
The second term in the first binomial is 4. We multiply 4 by each term in the second binomial:
4(2) = 8 4(3x) = 12x
Step 3: Combine Like Terms
Now that we have multiplied each term in the first binomial by each term in the second binomial, we can combine like terms. Like terms are terms that have the same variable raised to the same power.
In this case, we have the following like terms:
- 2x and 12x (both have the variable x raised to the power of 1)
- 3x^2 (has the variable x raised to the power of 2)
We can combine the like terms by adding their coefficients:
2x + 12x = 14x 3x^2 (no like terms to combine)
Simplifying the Expression
Now that we have combined like terms, we can simplify the expression by removing any unnecessary components. In this case, we have:
$(x+4)(2+3x)$ = 2x + 12x + 3x^2 + 8
Combining like terms, we get:
$(x+4)(2+3x)$ = 14x + 3x^2 + 8
Conclusion
In this article, we simplified the given expression $(x+4)(2+3x)$ by multiplying each term in the first binomial by each term in the second binomial and combining like terms. The simplified expression is:
$(x+4)(2+3x)$ = 14x + 3x^2 + 8
This expression is now in its simplest form, and we can use it to solve equations and inequalities.
Final Answer
The final answer is:
Simplify: {(x+4)(2+3x)$}$
In the previous article, we simplified the expression {(x+4)(2+3x)$}$. In this article, we will answer some common questions related to simplifying algebraic expressions.
Q: What is the distributive property?
A: The distributive property is a fundamental concept in algebra that allows us to multiply each term in one binomial by each term in another binomial. It states that for any real numbers a, b, c, and d:
a(b + c) = ab + ac
Q: How do I multiply binomials?
A: To multiply binomials, you need to follow these steps:
- Multiply each term in the first binomial by each term in the second binomial.
- Combine like terms.
- Simplify the expression.
For example, to multiply ($(x+4)$ and $(2+3x)$, you would follow these steps:
- Multiply x by 2 and x by 3x: 2x and 3x^2
- Multiply 4 by 2 and 4 by 3x: 8 and 12x
- Combine like terms: 2x + 12x = 14x, 3x^2 (no like terms to combine), and 8
Q: What are like terms?
A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 12x 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 their coefficients. For example, to combine 2x and 12x, you would add their coefficients:
2x + 12x = 14x
Q: What is the difference between a coefficient and a variable?
A: A coefficient is a number that is multiplied by a variable. For example, in the term 2x, 2 is the coefficient and x is the variable.
Q: Can I simplify an expression that has more than two binomials?
A: Yes, you can simplify an expression that has more than two binomials. You would follow the same steps as before:
- Multiply each term in the first binomial by each term in the second binomial.
- Multiply each term in the resulting expression by each term in the third binomial.
- Combine like terms.
- Simplify the expression.
Q: What are some common mistakes to avoid when simplifying expressions?
A: Some common mistakes to avoid when simplifying expressions include:
- Not combining like terms
- Not multiplying each term in one binomial by each term in another binomial
- Not simplifying the expression after combining like terms
Conclusion
In this article, we answered some common questions related to simplifying algebraic expressions. We covered topics such as the distributive property, multiplying binomials, like terms, combining like terms, coefficients, and variables. We also provided some tips on how to avoid common mistakes when simplifying expressions.
Final Answer
The final answer is: