Use the distributive property to show that each
a. $(x+2)(x+4)$ and $x^2+6 x+8$
Answer (2)
Expand ( x + 2 ) ( x + 4 ) using the distributive property.
Multiply first terms: x × x = x 2 .
Multiply outer, inner, and last terms: x × 4 = 4 x , 2 × x = 2 x , 2 × 4 = 8 .
Combine and simplify: x 2 + 4 x + 2 x + 8 = x 2 + 6 x + 8 . Therefore, ( x + 2 ) ( x + 4 ) = x 2 + 6 x + 8 . ( x + 2 ) ( x + 4 ) = x 2 + 6 x + 8
Explanation
Understanding the Problem We are asked to use the distributive property to show that ( x + 2 ) ( x + 4 ) is equivalent to x 2 + 6 x + 8 . This involves expanding the product ( x + 2 ) ( x + 4 ) and simplifying the resulting expression to see if it matches x 2 + 6 x + 8 .
Applying the Distributive Property To expand ( x + 2 ) ( x + 4 ) , we use the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This means we multiply each term in the first parentheses by each term in the second parentheses.
Multiplying First Terms First, we multiply the first terms in each parentheses: x × x = x 2 .
Multiplying Outer Terms Next, we multiply the outer terms: x × 4 = 4 x .
Multiplying Inner Terms Then, we multiply the inner terms: 2 × x = 2 x .
Multiplying Last Terms Finally, we multiply the last terms: 2 × 4 = 8 .
Combining the Terms Now, we combine all the terms we obtained: x 2 + 4 x + 2 x + 8 .
Simplifying the Expression We simplify the expression by combining like terms. The like terms are 4 x and 2 x , which add up to 6 x . So, the simplified expression is x 2 + 6 x + 8 .
Conclusion Since expanding ( x + 2 ) ( x + 4 ) and simplifying gives us x 2 + 6 x + 8 , we have shown that ( x + 2 ) ( x + 4 ) is indeed equivalent to x 2 + 6 x + 8 using the distributive property.
Examples
The distributive property is a fundamental concept in algebra and is used in many real-world applications. For example, suppose you are designing a rectangular garden. If you want to increase the length by 2 feet and the width by 4 feet from an initial size of 'x' feet for both length and width, the new area can be represented as ( x + 2 ) ( x + 4 ) . Expanding this using the distributive property gives x 2 + 6 x + 8 , which helps you calculate the total area of the expanded garden. This principle is also used in calculating areas of rooms, volumes of objects, and many other practical scenarios.
By applying the distributive property (or FOIL method) to the expression ( x + 2 ) ( x + 4 ) , we find that it simplifies to x 2 + 6 x + 8 . First, we calculate the products of each pair of terms from the binomials, combine like terms, and conclude with the equivalent expression. Thus, it shows that the two sides are equal as required.
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