91Ó°ÊÓ

The distributive property is a fundamental principle in algebra. It allows you to multiply a sum by multiplying each addend individually and then adding the products. This property can be especially useful in polynomial multiplication.
To apply the distributive property, you need to carefully multiply each term in one polynomial by each term in the other. In our example problem \( (4x+1)(2x+7) \) , we distribute the terms as follows:
1. Multiply the first term of the first polynomial \( (4x) \) by both terms in the second polynomial \(2x \) and \(7 \)
2. Repeat for the second term \(1 \) in the first polynomial. This gives us four separate products: \(4x \times 2x, 4x \times 7, 1 \times 2x, \) and \(1 \times 7 \).

After expanding the polynomials, the next step is to combine like terms. 'Like terms' are terms that have identical variable parts. For example, in \(8x^2 + 28x + 2x + 7 \), both \(28x \) and \(2x \) are like terms because they both contain the variable \(x \).
To combine like terms, simply add their coefficients. In our example:

• \text{Add the coefficients of \(28x \) and \(2x \):} \( 28 + 2 = 30 \)

So, the expression simplifies from \(8x^2 + 28x + 2x + 7 \) to \(8x^2 + 30x + 7 \). Combining like terms makes the expression more manageable and simpler to understand.

Binomial multiplication is a specific case of polynomial multiplication where each polynomial has exactly two terms. When multiplying binomials, the process can be summarized using the distributive property, focusing on each term sequentially.

Let's walk through the example \( (4x+1)(2x+7) \) step-by-step:

• Start with \(4x \): \(4x \times 2x = 8x^2 \) and \ (4x \times 7 = 28x) , so we get \( 8x^2 + 28x \).
• Apply the same process to \ ( 1) : \ (1 \times 2x = 2x \) and \(1 \times 7 = 7 \), resulting 2x + 7 \.

Finally, sum all the products: \(8x^2 + 28x + 2x + 7 \) and then combine like terms: \(8x^2 + 30x +7 \. This straightforward approach allows you to manage and simplify your polynomial expressions efficiently.