Multiplying a trinomial by another trinomial involves combining these three-term expressions to produce a new trinomial or a more complex polynomial. The entities involved in this process include the original trinomials, the coefficients of the terms within each trinomial, the constant terms, and the multiplication operation itself. By understanding these entities and their relationships, learners can master the technique of multiplying trinomials, a fundamental mathematical operation in algebra.
Multiplying a Trinomial by a Trinomial
Multiplying trinomials, expressions with three terms, can seem daunting. But by following a structured approach, you can simplify the process:
1. Set Up the Multiplication:
- Write out the first trinomial as the first row.
- Write out the second trinomial as the first column.
- Create a grid with as many rows and columns as the number of terms in each trinomial.
2. Multiply First Trinomial by Each Term of Second Trinomial:
- Multiply the first term of the first trinomial by all three terms of the second trinomial.
- Write the results in the first row of the grid.
- Repeat this step for each term of the first trinomial.
3. Multiply Each Term of First Trinomial by Second Trinomial:
- Multiply each term of the second trinomial by all three terms of the first trinomial.
- Write the results in the first column of the grid.
- Repeat this step for each term of the second trinomial.
4. Combine Like Terms:
- Look for terms with the same variable combinations (e.g., x²y³, x²y², and x²y).
- Combine their coefficients.
5. Simplify:
- Multiply all the constants.
- Combine like terms (e.g., 5x³ + 3x³ = 8x³)
- Write the final product.
Example:
Multiply: (2x – 3y + 1)(x² – 2y + 5)
1. Set Up:
2x - 3y + 1
x² - 2y + 5
2. Multiply First Trinomial by Each Term of Second Trinomial:
2x(x²) - 2x(2y) + 2x(5) = 2x³ - 4xy + 10x
-3y(x²) + 3y(2y) - 3y(5) = -3x²y + 6y² - 15y
1(x²) - 1(2y) + 1(5) = x² - 2y + 5
3. Multiply Each Term of First Trinomial by Second Trinomial:
x²(2x) - 2y(2x) + 5(2x) = 2x³ - 4xy + 10x
-2y(2x) + 2y(2y) - 5(2x) = -4xy + 4y² - 10y
x²(1) - 2y(1) + 5(1) = x² - 2y + 5
4. Combine Like Terms:
2x³ - 4xy + 10x - 4xy + 4y² - 10y - 3x²y + 6y² - 15y + x² - 2y + 5 = ...
5. Simplify:
2x³ - 3x²y + x² + 4y² + 4y - 4xy + 10x - 15y + 5
Question 1:
How is the multiplication of trinomials performed?
Answer:
Trinomial multiplication involves multiplying three binomials using the distributive property. Each term of the first binomial is multiplied by each term of the second binomial, and the products are then added together. The same process is repeated with the third binomial.
Question 2:
What is the FOIL method for multiplying trinomials?
Answer:
The FOIL method (First, Outer, Inner, Last) is a mnemonic device used to simplify trinomial multiplication. It involves multiplying the first, outer, inner, and last terms of the trinomials and adding the products together.
Question 3:
How can cross-multiplication be applied to trinomial multiplication?
Answer:
Cross-multiplication involves multiplying the outer terms and the inner terms of the two trinomials and adding the products together. This provides two of the four products required for the full trinomial multiplication.
Hey folks, thanks for hanging out with me while we tackled the wild world of trinomial multiplication. It’s been a mathematical adventure, but we made it through together. Remember, practice makes perfect, so keep on crunching those numbers and you’ll be a trinomial multiplier extraordinaire in no time. Now go forth and conquer all those pesky expressions! And hey, don’t be a stranger. Come back and visit again soon for more mathemagical fun.