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Learn it your way
A short class with Sarah and her students. The other ways are under More.
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Welcome everyone! Today, we're diving into multiplying polynomials. Can anyone tell me what a polynomial is?
Isn't a polynomial an expression made up of variables and coefficients?
Exactly, Student_1! Now, when we multiply polynomials, we often start with binomials. Let's look at how we can multiply them using the distributive law. Can anyone remind me what that law states?
It means we can distribute each term over the others!
Well done! For example, in multiplying the binomial (2a + 3b) with (3a + 4b), we would do this: multiply each term in the first binomial by each term in the second one.
So we would do 2a times 3a, then 2a times 4b, and so on?
Exactly right! Each multiplication gives us a term, and we then combine like terms.
What happens if there are no like terms?
Good question! If there are no like terms, we just list all the terms in our final answer. Now let’s do a quick example together.
Let's multiply (x - 4) and (2x + 3). Who wants to lead the way?
I’ll try! First, I do x times 2x, which is 2x².
Great start! Now what’s next?
Then I do x times 3, which gives me 3x.
Perfect! Now let’s move on to -4. What do we get?
That gives -8x and -12, right?
Correct again! Now let's combine those like terms to finish.
So, 2x² - 5x - 12 is our final answer?
Exactly! Great teamwork! Remember to always combine like terms when finishing up.
Now, let’s increase the complexity. How do you think we would multiply a binomial with a trinomial? For instance, (a + 7)(a² + 3a + 5).
Do we still use the distributive law?
That's right! We distribute each term in the binomial over each term in the trinomial.
So it’s like multiplying the binomial by each term in the trinomial?
Exactly! You will end up with three terms from the trinomial multiplied by two from the binomial. This means we’ll have six total terms to combine. Let’s work through it together!
I see! So, we would get a³, 3a², 5a, 7a², 21a, and 35.
Great job! Now can anyone tell me how we simplify those terms?
We combine like terms! So we’d get a³ + 10a² + 26a + 35.
Absolutely! Remembering to combine like terms is key in these multiplications.
A mathematical expression comprising variables and coefficients.
A polynomial with two terms.
A polynomial with three terms.
A fundamental property used to simplify multiplication.
The process of combining expressions with the same variables and exponents.
To multiply two binomials, you expand with glee, every term you see.
Imagine multiplying the prices of apples and oranges; combine them for a fruit basket total.
Remember 'FOIL' when multiplying binomials: First, Outside, Inside, Last.
BIP - Binomial Interaction Principle: remember to combine like terms!
Term
What do we call an expression with two terms?
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Term
What is the result of (x + 1)(x - 1)?
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An algebraic expression composed of variables and coefficients, involving only non-negative integer exponents.
A polynomial with exactly two terms.
A polynomial with exactly three terms.
A property that allows for the multiplication of a sum by distributing the multiplier across each addend.
Terms that contain the same variables raised to the same power.
This section covers the methods and processes involved in multiplying polynomials, specifically focusing on multiplying binomials and the distributive law.
In section 8.5, we learn about multiplying polynomials, particularly binomials. The process involves applying the distributive law, where each term in one polynomial multiplies every term in the other. Examples illustrate this process, emphasizing the importance of combining like terms to simplify the results.
In this section, we delve into the multiplication of polynomials, particularly focusing on the multiplication of binomials by binomials and binomials by trinomials using the distributive law.
Multiplication of Binomials: When multiplying two binomials, such as (2a + 3b) and (3a + 4b), every term in the first binomial is multiplied by every term in the second binomial. This process results in multiple terms, which may include like terms that can be combined for simplification. For example, the product (2a + 3b)(3a + 4b) yields several products: 6a² + 9ab + 8ab + 12b². Combining like terms here gives us 6a² + 17ab + 12b².
Distributive Law: The distributive law is pivotal in polynomial multiplication. It states that a(b + c) = ab + ac. This law allows us to expand the multiplication process systematically.
Practical Examples: Several examples demonstrate how to apply these principles, including how to simplify expressions and handle multiple terms.
Overall, mastering polynomial multiplication is essential for dealings in algebra, which often encounters expressions necessitating expansion, such as in area computation and in various algebraic identities.