Choose The Product.$\[ -7 P^3\left(4 P^2+3 P-1\right) \\]A. $\[ 28 P^6+21 P^3-7 P \\]B. $\[ -21 P^5+8 P^4-3 P^3 \\]C. $\[ -28 P^5-21 P^4+7 P^3 \\]D. $\[ 21 P^3+8 P^2+3 P^4-8 P \\]
Introduction
Multiplying polynomials is a fundamental concept in algebra that can seem daunting at first, but with practice and patience, it becomes a breeze. In this article, we will explore the process of multiplying polynomials, focusing on the distributive property and the order of operations. We will also provide examples and exercises to help you master this skill.
The Distributive Property
The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c:
a(b + c) = ab + ac
This property allows us to multiply a single term by a binomial (a sum of two terms). In the context of polynomials, the distributive property can be used to multiply a polynomial by a binomial.
Multiplying a Polynomial by a Binomial
To multiply a polynomial by a binomial, we can use the distributive property. Let's consider the following example:
-7p3(4p2 + 3p - 1)
To multiply this expression, we need to multiply each term in the polynomial by each term in the binomial.
Step 1: Multiply the first term in the polynomial by each term in the binomial
-7p3(4p2) = -28p^5 -7p^3(3p) = -21p^4 -7p^3(-1) = 7p^3
Step 2: Multiply the second term in the polynomial by each term in the binomial
-7p3(4p2) = -28p^5 -7p^3(3p) = -21p^4 -7p^3(-1) = 7p^3
Step 3: Multiply the third term in the polynomial by each term in the binomial
-7p3(4p2) = -28p^5 -7p^3(3p) = -21p^4 -7p^3(-1) = 7p^3
Step 4: Combine like terms
Now that we have multiplied each term in the polynomial by each term in the binomial, we can combine like terms.
-28p^5 - 21p^4 + 7p^3
Answer
The final answer is:
-28p^5 - 21p^4 + 7p^3
Conclusion
Multiplying polynomials is a fundamental concept in algebra that can seem daunting at first, but with practice and patience, it becomes a breeze. By using the distributive property and the order of operations, we can multiply polynomials with ease. Remember to multiply each term in the polynomial by each term in the binomial and combine like terms to get the final answer.
Exercises
- Multiply the following polynomial by the binomial:
2p^2(3p + 2)
- Multiply the following polynomial by the binomial:
3p^3(2p - 1)
- Multiply the following polynomial by the binomial:
4p^4(3p + 2)
Answer Key
- 6p^3 + 4p^2
- 6p^4 - 3p^3
- 12p^5 + 8p^4
Final Thoughts
Introduction
Multiplying polynomials is a fundamental concept in algebra that can seem daunting at first, but with practice and patience, it becomes a breeze. In this article, we will explore some common questions and answers related to multiplying polynomials.
Q: What is the distributive property?
A: The distributive property is a fundamental concept in algebra that states that for any real numbers a, b, and c:
a(b + c) = ab + ac
This property allows us to multiply a single term by a binomial (a sum of two terms).
Q: How do I multiply a polynomial by a binomial?
A: To multiply a polynomial by a binomial, we can use the distributive property. We need to multiply each term in the polynomial by each term in the binomial.
Q: What is the order of operations when multiplying polynomials?
A: When multiplying polynomials, we need to follow the order of operations:
- Multiply each term in the polynomial by each term in the binomial.
- Combine like terms.
Q: How do I combine like terms?
A: To combine like terms, we need to add or subtract the coefficients of the terms with the same variable and exponent.
Q: What is the difference between a polynomial and a binomial?
A: A polynomial is an expression consisting of one or more terms, each of which is a constant or a variable raised to a non-negative integer power. A binomial is a polynomial with two terms.
Q: Can I multiply a polynomial by a trinomial?
A: Yes, you can multiply a polynomial by a trinomial. To do this, you need to multiply each term in the polynomial by each term in the trinomial.
Q: How do I multiply a polynomial by a polynomial?
A: To multiply a polynomial by a polynomial, you need to multiply each term in the first polynomial by each term in the second polynomial.
Q: What is the difference between multiplying polynomials and multiplying expressions?
A: Multiplying polynomials involves multiplying expressions with variables and exponents, while multiplying expressions involves multiplying expressions with variables and constants.
Q: Can I use a calculator to multiply polynomials?
A: Yes, you can use a calculator to multiply polynomials. However, it's always a good idea to check your work by hand to make sure you get the correct answer.
Q: How do I check my work when multiplying polynomials?
A: To check your work, you can multiply the polynomial by a simple binomial, such as (x + 1), and see if the result is correct.
Conclusion
Multiplying polynomials is a fundamental concept in algebra that can seem daunting at first, but with practice and patience, it becomes a breeze. By following the distributive property and the order of operations, we can multiply polynomials with ease. Remember to combine like terms and check your work to get the correct answer.
Exercises
- Multiply the following polynomial by the binomial:
2p^2(3p + 2)
- Multiply the following polynomial by the binomial:
3p^3(2p - 1)
- Multiply the following polynomial by the binomial:
4p^4(3p + 2)
Answer Key
- 6p^3 + 4p^2
- 6p^4 - 3p^3
- 12p^5 + 8p^4
Final Thoughts
Multiplying polynomials is a fundamental concept in algebra that can seem daunting at first, but with practice and patience, it becomes a breeze. By following the distributive property and the order of operations, we can multiply polynomials with ease. Remember to combine like terms and check your work to get the correct answer. With practice and patience, you will become a master of multiplying polynomials in no time.