Problem 57 - Elementary Algebra

Question

Find the product: \((2 y+7)(3 y-1)\).

Step-by-Step Solution

Answer

Question: Find the product of the binomials (2y + 7)(3y - 1). Answer: The product of the binomials (2y + 7)(3y - 1) is 6y^2 + 19y - 7.

Step 1: Apply the distributive property

To find the product of the two binomials, we will multiply each term in the first binomial with each term in the second binomial. It will look like this: \((2y + 7)(3y - 1) = (2y)(3y) + (2y)(-1) + (7)(3y) + (7)(-1)\)

Step 2: Multiply the terms

Next, multiply the terms from the previous step: \((2y)(3y) = 6y^2\), \((2y)(-1) = -2y\), \((7)(3y) = 21y\), \((7)(-1) = -7\)

Step 3: Add and combine the like terms

Now, we will add and combine like terms (6y^2, -2y and 21y): \(6y^2 - 2y + 21y - 7\) Combining the like terms, we get: \(6y^2 + (-2y + 21y) - 7\)

Step 4: Simplify the expression

Simplify the expression by adding the like terms -2y and 21y: \(6y^2 + 19y - 7\) The product of \((2y + 7)(3y - 1)\) is \(6y^2 + 19y - 7\).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term inside a bracket. It is crucial when dealing with polynomial multiplication. This property states that, for any numbers or expressions \(a\), \(b\), and \(c\), the expression \(a(b + c)\) is equivalent to \(ab + ac\). This becomes especially helpful when multiplying binomials, which are polynomial expressions with two terms.

In our example, we applied the distributive property to multiply the binomials \((2y + 7)\) and \((3y - 1)\). To do this:

• Multiply \(2y\) by both \(3y\) and \(-1\).
• Multiply \(7\) by both \(3y\) and \(-1\).

This ensures that each term in the first binomial is distributed over each term in the second binomial, ultimately allowing us to expand and simplify the expression.

Binomials

A binomial is an algebraic expression that consists of exactly two terms. These terms are usually joined by either a plus or a minus sign. For example, the expressions \(2y + 7\) and \(3y - 1\) are both binomials. When multiplying binomials together, you must consider each combination of terms from both binomials to find the product.

Using our exercise as an example, the multiplication of binomials \((2y + 7)\) and \((3y - 1)\) requires us to:

• Multiply each term in the first binomial by each term in the second binomial.
• Apply the distributive property to systematically handle all cross-products.

Understanding how to properly handle binomial multiplication is a key skill in algebra, as it is frequently utilized in more complex mathematical problems.

Like Terms

Like terms in algebra are terms that have identical variables raised to the same power. Simplifying expressions by combining like terms is a common task when dealing with polynomial expressions. For example, in polynomial expressions like \(6y^2 - 2y + 21y - 7\), the terms \(-2y\) and \(21y\) are considered like terms because they both contain the variable \(y\) raised to the first power.

To simplify the expression, these like terms need to be combined:

• Add the coefficients of the like terms together.
• Keep the common variable and its power unchanged.

In our case, combining \(-2y\) and \(21y\) results in \(19y\). This reduction to like terms forms part of the final simplification, yielding the product from the multiplication of the original binomials.