Which polynomial represents the area of the rectangle? Responses 4x^(2)-2x-6 4 x squared minus 2 x minus 6 4x^(2)-6 4 x squared minus 6 4x^(2)-10x-6 4 x squared minus 10 x minus 6 4x^(2)+2x-6 4 x squared plus 2 x minus 6 Rectangle with one side labeled four x minus six and another side labeled x plus one
The Correct Answer and Explanation is:
To find the polynomial that represents the area of a rectangle, we use the formula:
$$
\text{Area} = \text{Length} \times \text{Width}
$$
In this case, the rectangle has one side labeled $4x – 6$ and the other side labeled $x + 1$. So we multiply these two expressions:
$$
(4x – 6)(x + 1)
$$
Now, apply the distributive property (also known as FOIL for binomials):
$$
(4x – 6)(x + 1) = 4x(x + 1) – 6(x + 1)
$$
Distribute each term:
$$
4x(x + 1) = 4x^2 + 4x
$$
$$
-6(x + 1) = -6x – 6
$$
Now add the results together:
$$
4x^2 + 4x – 6x – 6 = 4x^2 – 2x – 6
$$
Final Answer:
$$
\boxed{4x^2 – 2x – 6}
$$
To determine the area of a rectangle when its sides are given as algebraic expressions, we multiply the two expressions. The two expressions provided are $4x – 6$ and $x + 1$. These expressions represent the length and the width of the rectangle.
Multiplying binomials involves using the distributive property. This property tells us to multiply each term in the first expression by each term in the second.
Step 1: Multiply the first term of the first binomial, $4x$, by each term of the second binomial:
- $4x \cdot x = 4x^2$
- $4x \cdot 1 = 4x$
Step 2: Multiply the second term of the first binomial, $-6$, by each term of the second binomial:
- $-6 \cdot x = -6x$
- $-6 \cdot 1 = -6$
Step 3: Combine all terms:
- $4x^2 + 4x – 6x – 6$
Step 4: Combine like terms:
- $4x^2 – 2x – 6$
This polynomial, $4x^2 – 2x – 6$, represents the total area of the rectangle in terms of $x$. It accounts for all the changes in the side lengths as $x$ changes and is derived using fundamental algebraic principles. Therefore, the correct response is $4x^2 – 2x – 6$.
