Multiplying Polynomials
Description/Explanation/Highlights
Video Description
This video explains how to multiply together two polynomials.
Steps and Key Points to Remember
To add or subtract polynomials, follow these steps:
- Two polynomials written together such as \((4x+3)(2x-2)\) without an addition or subtraction sign between them implies multiplication.
- To multiply the two binomials in the above example together, distribute the first term in the first binomial to each term in the second and then distribute the second term in the first binomial to each term in the second. In this case, multiple \(4x\times2x=8x^2\) and then \(4x\times-2=-8x\). Now do the same thing with the \(3\) and multiply it by \(2x\) to get \(6x\) and by \(-2\) to get \(-6\). The resulting terms will be \(8x^2-8x+6x-6\) and when the like terms are combined, the final answer will be: \(8x^2-2x-6\).
- When multiplying a monomial by a polynomial, simply distribute the monomial to each term in the second polynomial. In the example \(2x(3x^2+x-4)\), multiply the monomial, \(2x\) by \(3x^2\), \(x\), and \(-4\). The resulting trinomial, \(6x^3+2x^2-8x\) is the final answer.
- The same rules apply when there are multiple variables. In the example \(2a^2b(3ab^2-4a-2b)\), multiply \(2a^2b\) by each term in the trinomial by multiplying coefficients and adding exponents. The result will be \(6a^3b^3-8a^3b-4a^2b^2\).
- To multiply larger polynomials such as a trinomial times a trinomial, follow the same rule and distribute each term in the first by each term in the second. Multiplication of a trinomial by a trinomial such as \(x^2+2xy+y^2)(2x^2-3x-4y)\), will result in 9 terms before combining like terms! In this example, the following will result from the multiplication: \(2x^4-3x^3-4x^2y+4x^3y-6x^2y-8xy^2+2x^2y^2-3xy^2-4y^3\). Combining like terms will result in a final answer of \(2x^4-3x^3-10x^2y+4x^3y-11xy^2+2x^2y^2-4y^3\).
Here are some key points to keep in mind multiplying polynomials.
- When two polynomials are surrounded by parenthesis and written together, multiplication is indicated.
- To multiply polynomials of any size, distribute by multiplying all terms in the first polynomial by all terms in the second polynomial.
- Remember, apply rules of exponents to variables with exponents–multiply coefficients and add exponents of the same variables (like bases).
- The number of terms in the first polynomial times the number of terms in the second polynomial will tell you how many terms the resulting multiplication will have before combining any like terms. (i.e. a trinomial [3 terms] multiplied by a trinomial will result in 9 terms before combining).
- Finish the problem by combining like terms. Be caseous and make sure variables and exponents in terms are exactly alike before combining.
Video Highlights
- 00:00 Introduction
- 00:28 Steps for multiplying polynomials using \((4x+3)(2x-2)\) as an example
- 02:25 \(2x(3x^2+x-4)\) monomial multiplied by a trinomial example
- 03:25 \(2a^2b(3ab^2-4a-2b)\) monomial multiplied by a trinomial example with multiple variables
- 04:28 \((x^2+2xy+y^2)(2x^2-3x-4y)\) trinomial times trinomial example
- 09:00 Conclusion
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