Question

The area, \(A(x),\) of a rectangle is represented by the polynomial \(2 x^{2}-x-6\) a) If the height of the rectangle is \(x-2\) what is the width in terms of \(x ?\) b) If the height of the rectangle were changed to \(x-3,\) what would the remainder of the quotient be? What does this remainder represent?

Short answer

a) Width is \(2x + 3\). b) Remainder is 9, representing additional area.

Step-by-step solution

Understanding the Problem

The area of the rectangle is given by the polynomial function: \[A(x) = 2x^2 - x - 6\] We are given the height of the rectangle is \(x-2\).

Finding the Width of the Rectangle

The area of a rectangle is given by the product of its height and width. So we can write: \[A(x) = \text{Height} \times \text{Width}\] Substituting the values we have: \[2x^2 - x - 6 = (x-2) \times \text{Width}\] To find the width, we need to divide the polynomial \(2x^2 - x - 6\) by \(x-2\).

Polynomial Division

Divide \(2x^2 - x - 6\) by \(x - 2\) using polynomial long division or synthetic division:1. Divide the leading term \(2x^2\) by \(x\) to get \(2x\).2. Multiply \(2x\) by \(x - 2\) to get \(2x^2 - 4x\).3. Subtract \(2x^2 - 4x\) from \(2x^2 - x - 6\) to get \(3x - 6\).4. Divide \(3x\) by \(x\) to get \(3\).5. Multiply \(3\) by \(x - 2\) to get \(3x - 6\).6. Subtract \(3x - 6\) from \(3x - 6\) to get \(0\).So, the width is \(2x + 3\).

Changing the Height

Now, change the height to \(x-3\) and find the remainder of the new quotient. We need to divide the polynomial \(2x^2 - x - 6\) by \(x - 3\).

Polynomial Division for New Height

Using synthetic division with \(x - 3\):1. The coefficients of \(2x^2 - x - 6\) are \(2, -1, -6\).2. Using the root \(3\):- Bring down the 2.- Multiply 2 by 3 to get 6.- Add -1 and 6 to get 5.- Multiply 5 by 3 to get 15.- Add -6 and 15 to get 9.The remainder is \(9\).

Interpreting the Remainder

The remainder, 9, represents the value that is left over and does not fit into the factor \(x-3\). In terms of the area, it means that there is an additional area that cannot be described by height \(x-3\).

Key concepts

Area of a Rectangle

The area of a rectangle is the amount of space inside the rectangle. It is calculated by multiplying the height by the width of the rectangle. In general terms, the area, A, can be given as:
$$ A = \text{Height} \times \text{Width} $$
In the original exercise, the area is given by the polynomial function:
$$ A(x) = 2x^2 - x - 6 $$ This means that, for any value of x, the area of the rectangle can be determined using this function. When the height is given as a polynomial, we can find the width by dividing the area polynomial by the height polynomial.

Polynomial Division

Polynomial division is similar to long division with numbers, but it involves dividing terms with variables. The goal is to divide one polynomial by another to find the quotient and remainder. In the exercise, we need to find the width of the rectangle by dividing the area polynomial by the height polynomial.
Steps involved in polynomial division:

• Identify the leading terms.
• Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this term and subtract from the dividend to get a new polynomial.
• Repeat the process with the new polynomial until no terms remain or the degree of the new polynomial is less than the divisor.

In the given problem, dividing the polynomial \(2x^2 - x - 6\) by \(x - 2\), we find that the width is \(2x + 3\).

Remainders in Division

When performing polynomial division, the remainder is what is left after dividing. It is similar to the remainder we get when dividing integers. In the original exercise, changing the height of the rectangle meant dividing the polynomial \(2x^2 - x - 6\) by \(x - 3\). The steps for synthetic division led us to a remainder of 9.
What does this remainder represent in the context of our problem?

• The remainder indicates that there is an additional area outside what can be accounted for by the new height \(x - 3\).

To summarize, remainders in polynomial division show us the extra 'stuff' that doesn't fit perfectly into our division factor. It highlights areas or values not divisible cleanly by our divisor.