Multiplying and Dividing Monomials
Multiplying a Polynomial and a Monomial
Here is an inline formula: \(e^{i\pi} + 1 = 0\)
Here is a displayed equation:
$$
\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}
$$
Here is a displayed equation:
$$
\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}
$$
For multiplying a polynomial with a monomial, we should consider two concepts in mind: multiplying monomial and distributive property.
We already are familiar with multiplying monomials.
The next concept is the distributive property, one of the most important properties in algebra.
When multiplying a monomial by a polynomial, we can multiply the monomial by each term in the polynomial. Here are the general forms:
$$a \times (b + c) = a \times b + a \times c = ab + ac,$$
$$a \times (b – c) = a \times b – a \times c = ab – ac.$$
Key Point
The distributive property: When you multiply a polynomial with a monomial, you multiply the monomial by each term in the polynomial individually,
rather than to the polynomial as a whole.
To make these concepts firm, let us consider some examples.
Example
Multiply the expressions \(6x(2x + 5).\)
Solution
Apply the distributive property:
\(6x(2x + 5) = 6x \times 2x + 6x \times 5 = 12x^{2} + 30x.\)
Example
Multiply the expressions \(x\left( 3x^{2} + 4y^{2} \right).\)
Solution
Using the distributive property:
\(x\left( 3x^{2} + 4y^{2} \right) = x \times 3x^{2} + x \times 4y^{2} = 3x^{3} + 4xy^{2}.\)
In step by step fashion, follow this:
To multiply binomials, apply these steps:
Multiply the first terms of both binomials (First).
Multiply the outer terms of both binomials (Outer).
Multiply the inner terms of both binomials (Inner).
Multiply the last terms of both binomials (Last).
Combine like terms if necessary.
Let’s use steps in an example:
Example
Multiply Binomials \((x – 5)(x + 4).\)
Solution22
The first terms are \(x\) and \(x\), so \(x \times x = x^{2}\).
The outer terms are \(x\) and \(4\), so \(x \times 4 = 4x\).
The inner terms are \(-5\) and \(x\), so \(-5 \times x = -5x\).
The last terms are \(-5\) and \(4\), so \(-5 \times 4 = -20\).
Combine like terms: \(x^{2} + 4x – 5x – 20 = x^{2} – x – 20\).
Formula to Remember for Factoring Trinomials
Reverse FOIL Method
Difference of Squares
Perfect Square Trinomial (Sum)
Perfect Square Trinomial (Difference)
Practices for Find each product:
\( 3x(x – 2) = \)
\( 5y(2y^2 + 3x) = \)
\( 4x(3x^2 – y) = \)
\( 2y(y^2 + 4x^2) = \)
\( 3x(4x^3 – 2y^2) = \)
Answer keys for Multiplying a Polynomial and a Monomial
\( 3x^2 – 6x \)
\( 10y^3 + 15xy \)
\( 12x^3 – 4xy \)
\( 2y^3 + 8yx^2 \)
\( 12x^4 – 6xy^2 \)
Explained answers for Multiplying a Polynomial and a Monomial
Apply the distributive property: \( 3x \times x + 3x \times -2 = 3x^2 – 6x \).
Apply the distributive property: \( 5y \times 2y^2 + 5y \times 3x = 10y^3 + 15xy \).
Apply the distributive property: \( 4x \times 3x^2 + 4x \times -y = 12x^3 – 4xy \).
Apply the distributive property: \( 2y \times y^2 + 2y \times 4x^2 = 2y^3 + 8yx^2 \).
Apply the distributive property: \( 3x \times 4x^3 + 3x \times -2y^2 = 12x^4 – 6xy^2 \).
What people say about " » Multiplying and Dividing Monomials"?
No one replied yet.