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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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Today we're going to discuss how to multiply a monomial by a polynomial. Remember, a monomial is an expression with just one term. Can anyone give me an example of a monomial?
4x is a monomial!
That's correct! Now, a polynomial can have one or more terms. Who can give me an example?
How about x² - 3x + 5?
Great job! Now, we will focus on multiplying a monomial with a polynomial, such as the binomial. What do we remember about the distributive law?
It means we multiply each term in the parentheses separately!
Exactly! Let's remember to distribute with an acronym: D for Distribute, E for Every term. Now, let’s see an example.
Let’s compute 3x × (5y + 2). Using the distributive law: what will be our first step?
We multiply 3x by both 5y and 2!
That's right! So we get: 15xy + 6x. Can anyone simplify: 7x multiplied by (2a - 4)?
That would be 14ax - 28!
Fantastic! Let's summarize: Remember to always apply our D.E. method. Distribute across every term.
Next, let’s multiply a monomial by a trinomial. For instance, 3p × (4p² + 5p + 7). What’s our first step?
We’ll multiply 3p by each term in the trinomial!
Correct! So, we get 12p³ + 15p² + 21p. Why do we have three terms?
Because there are three terms in the trinomial!
Exactly! Great job! Remember to look out for combining like terms when multiplying!
To apply these skills, let’s think of practical examples. If I say the length of a rectangle is (3x) and its width is (4y + 2), what is the area?
We would multiply 3x by (4y + 2) to get 12xy + 6x!
Great! Let's practice multiplying some more. What is the product of 5(a + 3)?
It's 5a + 15.
Fantastic! I want everyone to practice these problems and remember our D.E. method.
A method for multiplying a single term by each term in a polynomial.
An algebraic expression consisting of exactly one term.
An algebraic expression containing one or more terms.
A polynomial with two terms.
A polynomial with three terms.
When you multiply, don't forget the rule, Distribute each term, that's the golden tool!
Once upon a time, there was a wise owl named 'Distributive.' He taught the students to share everything equally, so when multiplying, they learned to distribute their values across all terms in a polynomial.
D.E. = Distribute Every term! A reminder to always distribute correctly!
M.P.P. = Monomial x Polynomial = Product in Polynomial form.
Term
Distributive Law
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Term
Monomial
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Term
Binomial
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Term
Trinomial
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This section focuses on the multiplication of monomials by polynomials, showing how to apply the distributive law effectively.
In this section, students learn how to multiply a monomial by a polynomial, covering both binomials and trinomials. The distributive law is introduced as a fundamental concept to simplify calculations, and examples illustrate the step-by-step process and applications.
This section, titled "Multiplying a Monomial by a Polynomial," outlines the method of multiplying monomials with polynomials, highlighting binomials and trinomials as key categories of polynomial expressions.
The multiplication of a monomial by a binomial is illustrated using examples, such as multiplying 3x by the binomial (5y + 2). Using the distributive law, the multiplication is expressed as:
The section further explores multiplying a monomial by a trinomial through an example of 3p × (4p² + 5p + 7), resulting in:
Prominent throughout is the emphasis on the distributive law, a principle that allows the simplification of expressions by distributing the monomial across each term in the polynomial. The section also underscores the importance of recognizing like terms when combining results after multiplication. By the end of this section, students will have established a clear understanding of multiplying monomials by polynomials, thereby strengthening their foundation for more complex algebraic operations.