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Chapter 3: Problem 11

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

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

Step by step solution

01

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 \(x-2\).
02

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 \(x-2\).
03

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 \(2x + 3\).
04

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 \(x - 3\).
05

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 \(9\).
06

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 \(x-3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

a) Divide the polynomial $$3 x^{4}-4 x^{3}-6 x^{2}+17 x-8 \text { by } x+1$$ using long division. Express the result in the form \(\frac{P(x)}{x-a}=Q(x)+\frac{R}{x-a}\) b) Identify any restrictions on the variable. c) Write the corresponding statement that can be used to check the division. d) Verify your answer.

Determine the values of \(m\) and \(n\) so that the polynomials \(2 x^{3}+m x^{2}+n x-3\) and \(x^{3}-3 m x^{2}+2 n x+4\) are both divisible by \(x-2\)

a) Given the function \(y=x^{3},\) list the parameters of the transformed polynomial function \(y=\left(\frac{1}{2}(x-2)\right)^{3}-3\) b) Describe how each parameter in part a) transforms the graph of the function \(y=x^{3}\) c) Determine the domain and range for the transformed function.

Without using technology, sketch the graph of each function. Label all intercepts. a) \(f(x)=x^{4}-4 x^{3}+x^{2}+6 x\) b) \(y=x^{3}+3 x^{2}-6 x-8\) c) \(y=x^{3}-4 x^{2}+x+6\) d) \(h(x)=-x^{3}+5 x^{2}-7 x+3\) e) \(g(x)=(x-1)(x+2)^{2}(x+3)^{2}\) f) \(f(x)=-x^{4}-2 x^{3}+3 x^{2}+4 x-4\)

A medical researcher establishes that a patient's reaction time, \(r\), in minutes, to a dose of a particular drug is \(r(d)=-3 d^{3}+3 d^{2},\) where \(d\) is the amount of the drug, in millilitres, that is absorbed into the patient's blood. a) What type of polynomial function is \(r(d) ?\) b) What are the leading coefficient and constant of this function? c) Make a sketch of what you think the function will look like. Then, graph the function using technology. How does it compare to your sketch? d) What are the restrictions on the domain of this function? Explain why you selected those restrictions.
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