Algebraic Multiplication and Simplification of Binomials

Multiplying Binomials Using the Grid or Area Model

The objective of this mathematical exercise is to determine the product of two binomials and simplify the resulting expression. The specific problem addressed involves the expression $(5y - 3)(5y + 2)$. To solve this, a structural approach utilizing a grid or area model is employed to ensure each term of the first binomial is multiplied by every term of the second binomial.

In this model, the terms of the first binomial, $5y$ and $-3$, are placed along the vertical axis of the grid. The terms of the second binomial, $5y$ and $2$, are placed along the horizontal axis. The multiplication process proceeds by filling in each cell of the grid with the product of its corresponding row and column headers. First, multiplying the variable terms gives $5y \times 5y$, which equals $25y^{2}$. Next, multiplying the header of the first row by the second column provides $5y \times 2$, yielding $10y$. Multiplying the header of the second row by the first column provides $-3 \times 5y$, yielding $-15y$. Finally, multiplying the constant terms in the bottom-right corner, $-3 \times 2$, yields $-6$.

Once the individual products are calculated, they are combined into a single polynomial expression: $25y^{2} - 15y + 10y - 6$. The final step in the process is simplification, which involves combining like terms. The linear terms $-15y$ and $10y$ are combined by adding their coefficients, resulting in $-5y$. Consequently, the final simplified product is identified as $25y^{2} - 5y - 6$.

Multiplying Binomials Using the FOIL Method

Another fundamental algebraic task involves finding the product and simplifying the expression $(-2b + 5)(-b - 1)$. This calculation is performed using the FOIL method, an acronym designating the multiplication of the First, Outer, Inner, and Last terms of the binomials.

The process begins with the "First" terms: multiplying $-2b$ by $-b$. Applying the rule that the product of two negative numbers is positive, the result is $2b^{2}$. The next step involves the "Outer" terms: multiplying the term at the beginning of the first binomial, $-2b$, by the term at the end of the second binomial, $-1$. This results in a product of $2b$ (again, negative times negative equals positive). Subsequent to this, the "Inner" terms are multiplied: $5$ multiplied by $-b$ results in $-5b$. Finally, the "Last" terms, which are the constant terms $5$ and $-1$, are multiplied to give $-5$.

The full intermediate expression derived from these four steps is written as $2b^{2} + 2b - 5b - 5$. To simplify this expression to its final form, the like terms $2b$ and $-5b$ must be combined. Calculating the sum of these terms involves adding the coefficients $2$ and $-5$, which results in $-3b$. By combining all components, the final simplified result of the multiplication is $2b^{2} - 3b - 5$.