What Is The Product Of The Polynomials Below?$(7x + 7)(x + 2$\]A. $7x^2 + 9x + 7$ B. $7x^2 + 21x + 14$ C. $7x^2 + 21x + 28$ D. $7x^2 + 9x + 14$
Understanding the Problem
To find the product of the given polynomials, we need to apply the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. In this case, we have two binomials: (7x + 7) and (x + 2). We will multiply each term of the first binomial by each term of the second binomial.
Multiplying the Binomials
To find the product, we will multiply each term of the first binomial by each term of the second binomial.
Multiplying 7x by x
The first term of the first binomial is 7x, and the first term of the second binomial is x. Multiplying these two terms, we get:
7x * x = 7x^2
Multiplying 7x by 2
The first term of the first binomial is 7x, and the second term of the second binomial is 2. Multiplying these two terms, we get:
7x * 2 = 14x
Multiplying 7 by x
The second term of the first binomial is 7, and the first term of the second binomial is x. Multiplying these two terms, we get:
7 * x = 7x
Multiplying 7 by 2
The second term of the first binomial is 7, and the second term of the second binomial is 2. Multiplying these two terms, we get:
7 * 2 = 14
Combining the Terms
Now that we have multiplied each term of the first binomial by each term of the second binomial, we can combine the terms to find the product.
7x^2 + 14x + 7x + 14
Simplifying the Expression
We can simplify the expression by combining like terms. The terms 14x and 7x are like terms, so we can combine them to get:
7x^2 + 21x + 14
Conclusion
The product of the polynomials (7x + 7) and (x + 2) is 7x^2 + 21x + 14.
Answer
The correct answer is B. .
Discussion
This problem requires the application of the distributive property to find the product of two binomials. The distributive property states that for any real numbers a, b, and c, a(b + c) = ab + ac. In this case, we have two binomials: (7x + 7) and (x + 2). We multiplied each term of the first binomial by each term of the second binomial and then combined the terms to find the product.
Tips and Tricks
- When multiplying binomials, it is helpful to use the distributive property to break down the multiplication into smaller steps.
- Make sure to combine like terms when simplifying the expression.
- Check your answer by plugging in values for x to see if the expression is true.
Related Problems
- Find the product of the polynomials (x + 3) and (x + 5).
- Find the product of the polynomials (2x + 1) and (x - 2).
- Find the product of the polynomials (x - 4) and (x + 6).
Conclusion
In this article, we discussed how to find the product of two binomials using the distributive property. We applied the distributive property to find the product of the polynomials (7x + 7) and (x + 2) and simplified the expression to get the final answer. We also provided tips and tricks for solving similar problems and related problems for further practice.
Q: What is the distributive property, and how is it used in multiplying polynomials?
A: The distributive property is a mathematical concept that states that for any real numbers a, b, and c, a(b + c) = ab + ac. In the context of multiplying polynomials, the distributive property is used to break down the multiplication into smaller steps. This involves multiplying each term of the first polynomial by each term of the second polynomial.
Q: How do I multiply two binomials using the distributive property?
A: To multiply two binomials using the distributive property, follow these steps:
- Multiply each term of the first binomial by each term of the second binomial.
- Combine like terms to simplify the expression.
Q: What are like terms, and how do I combine them?
A: Like terms are terms that have the same variable and exponent. For example, 2x and 4x are like terms because they both have the variable x and the exponent 1. To combine like terms, add or subtract the coefficients of the like terms.
Q: How do I simplify an expression after multiplying polynomials?
A: To simplify an expression after multiplying polynomials, combine like terms and eliminate any unnecessary terms.
Q: What are some common mistakes to avoid when multiplying polynomials?
A: Some common mistakes to avoid when multiplying polynomials include:
- Failing to distribute the terms correctly
- Not combining like terms
- Not simplifying the expression after multiplying
Q: How do I check my answer when multiplying polynomials?
A: To check your answer when multiplying polynomials, plug in values for the variable and see if the expression is true. For example, if you are multiplying (x + 2) and (x + 3), plug in x = 0 and see if the expression equals 6.
Q: What are some real-world applications of multiplying polynomials?
A: Multiplying polynomials has many real-world applications, including:
- Calculating the area of a rectangle or triangle
- Finding the volume of a box or cylinder
- Modeling population growth or decline
- Solving systems of equations
Q: How do I multiply polynomials with negative coefficients?
A: To multiply polynomials with negative coefficients, follow the same steps as multiplying polynomials with positive coefficients. The only difference is that you will have negative terms in the final expression.
Q: Can I use a calculator to multiply polynomials?
A: Yes, you can use a calculator to multiply polynomials. However, it is still important to understand the concept of multiplying polynomials and to be able to do it by hand.
Q: How do I multiply polynomials with fractional coefficients?
A: To multiply polynomials with fractional coefficients, follow the same steps as multiplying polynomials with integer coefficients. The only difference is that you will have fractional terms in the final expression.
Q: What are some tips for multiplying polynomials quickly and accurately?
A: Some tips for multiplying polynomials quickly and accurately include:
- Using the distributive property to break down the multiplication into smaller steps
- Combining like terms as you go
- Checking your answer by plugging in values for the variable
- Practicing regularly to build your skills and confidence.