Write a polynomial that represents the length of the rectangle.
The width is x+0.2 ?units.
The area is 0.8×3+0.56×2+0.58x+0.1 ?square units.
The length is units.
(Use integers or decimals for any numbers in the expression.)
The Correct Answer and Explanation is:
To find the length of the rectangle, we use the relationship between area, length, and width of a rectangle: Area=Length×Width\text{Area} = \text{Length} \times \text{Width}
We are given:
- Width = x+0.2x + 0.2
- Area = 0.8×3+0.56×2+0.58x+0.10.8x^3 + 0.56x^2 + 0.58x + 0.1
We need to solve for length, so we rearrange the formula: Length=AreaWidth=0.8×3+0.56×2+0.58x+0.1x+0.2\text{Length} = \frac{\text{Area}}{\text{Width}} = \frac{0.8x^3 + 0.56x^2 + 0.58x + 0.1}{x + 0.2}
Step-by-step Polynomial Division:
We’ll divide the cubic polynomial by the linear binomial using polynomial long division:
Divide:
0.8×3÷x=0.8×20.8x^3 \div x = 0.8x^2
Multiply and Subtract: (x+0.2)(0.8×2)=0.8×3+0.16×2(x + 0.2)(0.8x^2) = 0.8x^3 + 0.16x^2
Subtract: (0.8×3+0.56×2)−(0.8×3+0.16×2)=0.4×2(0.8x^3 + 0.56x^2) – (0.8x^3 + 0.16x^2) = 0.4x^2
Bring down: +0.58x
Next term:
0.4×2÷x=0.4×0.4x^2 \div x = 0.4x
Multiply:
(x+0.2)(0.4x)=0.4×2+0.08x(x + 0.2)(0.4x) = 0.4x^2 + 0.08x
Subtract: (0.4×2+0.58x)−(0.4×2+0.08x)=0.5x(0.4x^2 + 0.58x) – (0.4x^2 + 0.08x) = 0.5x
Bring down: +0.1
Next term:
0.5x÷x=0.50.5x \div x = 0.5
Multiply:
(x+0.2)(0.5)=0.5x+0.1(x + 0.2)(0.5) = 0.5x + 0.1
Subtract: (0.5x+0.1)−(0.5x+0.1)=0(0.5x + 0.1) – (0.5x + 0.1) = 0
So, there is no remainder.
Final Answer:
Length=0.8×2+0.4x+0.5\text{Length} = 0.8x^2 + 0.4x + 0.5
Explanation (300+ words):
To determine the length of the rectangle when the area and width are known, we use a fundamental geometric formula:
Area = Length × Width.
Rearranging the formula, we isolate Length: Length=AreaWidth\text{Length} = \frac{\text{Area}}{\text{Width}}
Here, the area is given as a cubic polynomial: 0.8×3+0.56×2+0.58x+0.10.8x^3 + 0.56x^2 + 0.58x + 0.1
And the width is a binomial expression: x+0.2x + 0.2
To find the quotient (which represents the length), we divide the area polynomial by the width polynomial. This is done using polynomial division. In this method, we divide the highest-degree term in the numerator by the highest-degree term in the denominator to determine the first term of the quotient. Then we multiply and subtract successively until we reduce the polynomial or reach a remainder of zero.
In this case, the polynomial division yields: 0.8×2+0.4x+0.50.8x^2 + 0.4x + 0.5
This expression is the length of the rectangle, as multiplying it by the width x+0.2x + 0.2 would regenerate the original area polynomial. Also, since the division resulted in no remainder, this confirms that the width is a true factor of the area, and the result is a valid expression for length.
This process of dividing one polynomial by another is crucial in algebra for simplifying expressions and understanding relationships among geometric dimensions expressed algebraically.