Multiply The Following Expressions:$\[ \begin{array}{r} 3x^2 - 7x + 4 \\ \times \quad (5x^2 + 2x) \\ \hline \end{array} \\]Choose The Correct Result From The Options Below:A. \[$[15x^4 - 35x^3 + 6x^2 + 8x]\$\]B. \[$[15x^4 - 29x^3 +
Introduction
In algebra, multiplying expressions is a fundamental operation that helps us simplify complex equations and solve problems. In this article, we will focus on multiplying two quadratic expressions, which is a common scenario in mathematics. We will use the distributive property to multiply the expressions and then simplify the result.
The Distributive Property
The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms. It states that for any numbers a, b, and c:
a(b + c) = ab + ac
This property can be extended to more complex expressions, including quadratic expressions.
Multiplying Quadratic Expressions
To multiply two quadratic expressions, we can use the distributive property to multiply each term in the first expression by each term in the second expression. Let's consider the following example:
(3x^2 - 7x + 4) × (5x^2 + 2x)
We will multiply each term in the first expression by each term in the second expression and then combine like terms.
Step 1: Multiply the First Term
The first term in the first expression is 3x^2. We will multiply this term by each term in the second expression:
3x^2 × 5x^2 = 15x^4 3x^2 × 2x = 6x^3
Step 2: Multiply the Second Term
The second term in the first expression is -7x. We will multiply this term by each term in the second expression:
-7x × 5x^2 = -35x^3 -7x × 2x = -14x^2
Step 3: Multiply the Third Term
The third term in the first expression is 4. We will multiply this term by each term in the second expression:
4 × 5x^2 = 20x^2 4 × 2x = 8x
Step 4: Combine Like Terms
Now that we have multiplied each term in the first expression by each term in the second expression, we can combine like terms:
15x^4 - 35x^3 + 6x^3 - 14x^2 + 20x^2 + 8x
Combining like terms, we get:
15x^4 - 29x^3 + 6x^2 + 8x
Conclusion
In this article, we have multiplied two quadratic expressions using the distributive property. We have broken down the multiplication process into four steps and combined like terms to simplify the result. The correct result is:
15x^4 - 29x^3 + 6x^2 + 8x
This is option A in the given choices.
Discussion
The multiplication of quadratic expressions is a fundamental operation in algebra that helps us simplify complex equations and solve problems. In this article, we have used the distributive property to multiply two quadratic expressions and then combined like terms to simplify the result. This process can be extended to more complex expressions and is an essential skill for any mathematician.
Common Mistakes
When multiplying quadratic expressions, it is easy to make mistakes. Some common mistakes include:
- Forgetting to multiply each term in the first expression by each term in the second expression
- Not combining like terms correctly
- Making errors when multiplying or combining terms
To avoid these mistakes, it is essential to follow the distributive property carefully and to check your work regularly.
Real-World Applications
The multiplication of quadratic expressions has many real-world applications. For example:
- In physics, quadratic expressions are used to model the motion of objects under the influence of gravity or other forces.
- In engineering, quadratic expressions are used to design and optimize systems, such as bridges or buildings.
- In economics, quadratic expressions are used to model the behavior of markets and economies.
Introduction
In our previous article, we discussed the process of multiplying quadratic expressions using the distributive property. In this article, we will answer some common questions related to multiplying expressions.
Q: What is the distributive property?
A: The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms. It states that for any numbers a, b, and c:
a(b + c) = ab + ac
This property can be extended to more complex expressions, including quadratic expressions.
Q: How do I multiply two quadratic expressions?
A: To multiply two quadratic expressions, we can use the distributive property to multiply each term in the first expression by each term in the second expression. Let's consider the following example:
(3x^2 - 7x + 4) × (5x^2 + 2x)
We will multiply each term in the first expression by each term in the second expression and then combine like terms.
Q: What is the difference between multiplying and adding expressions?
A: When we multiply expressions, we are multiplying each term in one expression by each term in the other expression. When we add expressions, we are combining like terms.
For example:
(3x^2 - 7x + 4) + (5x^2 + 2x) = 8x^2 - 5x + 4
(3x^2 - 7x + 4) × (5x^2 + 2x) = 15x^4 - 29x^3 + 6x^2 + 8x
Q: How do I simplify a product of two expressions?
A: To simplify a product of two expressions, we can use the distributive property to multiply each term in the first expression by each term in the second expression and then combine like terms.
For example:
(3x^2 - 7x + 4) × (5x^2 + 2x) = 15x^4 - 29x^3 + 6x^2 + 8x
Q: What are some common mistakes to avoid when multiplying expressions?
A: Some common mistakes to avoid when multiplying expressions include:
- Forgetting to multiply each term in the first expression by each term in the second expression
- Not combining like terms correctly
- Making errors when multiplying or combining terms
To avoid these mistakes, it is essential to follow the distributive property carefully and to check your work regularly.
Q: How do I use the distributive property to multiply a binomial by a trinomial?
A: To multiply a binomial by a trinomial, we can use the distributive property to multiply each term in the binomial by each term in the trinomial.
For example:
(2x + 3) × (x^2 + 2x + 1) = 2x^3 + 4x^2 + 2x + 3x^2 + 6x + 3
We will multiply each term in the binomial by each term in the trinomial and then combine like terms.
Q: How do I use the distributive property to multiply a polynomial by a monomial?
A: To multiply a polynomial by a monomial, we can use the distributive property to multiply each term in the polynomial by the monomial.
For example:
(2x^2 + 3x + 1) × 4 = 8x^2 + 12x + 4
We will multiply each term in the polynomial by the monomial and then combine like terms.
Conclusion
In this article, we have answered some common questions related to multiplying expressions. We have discussed the distributive property, how to multiply two quadratic expressions, and how to simplify a product of two expressions. We have also covered some common mistakes to avoid when multiplying expressions and how to use the distributive property to multiply a binomial by a trinomial and a polynomial by a monomial.