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Chapter 3: Problem 17

Write a polynomial that satisfies each set of conditions. a) a quadratic polynomial that gives a remainder of -4 when it is divided by \(x-3\) b) a cubic polynomial that gives a remainder of 4 when it is divided by \(x+2\) c) a quartic polynomial that gives a remainder of 1 when it is divided by \(2 x-1\)

Short Answer

Expert verified
P(x) = x^2 + x - 16; Q(x) = x^3 + 2x^2 + x + 6; R(x) = x^4 + 2x^3 + 3x^2 + 4x - 17/8.

Step by step solution

01

- Understand the Remainder Theorem

The Remainder Theorem states that if a polynomial function f(x) is divided by x-c, the remainder of that division is f(c). For a polynomial divided by ax-b, the remainder is f(b/a).
02

- Formulate the Quadratic Polynomial

For the quadratic polynomial P(x), it must give a remainder of -4 when divided by x-3. According to the Remainder Theorem, P(3) = -4. Let P(x) = ax^2 + bx + c. Substitute x = 3 into P(x) to get: \( a(3)^2 + b(3) + c = -4 \). This simplifies to: \( 9a + 3b + c = -4 \). Since we don't have more conditions, we assume simplest form: Let a=1, b=1 and solve for c: \( 9(1) + 3(1) + c = -4 \), resulting in c = -16. So, P(x) = x^2 + x - 16.
03

- Formulate the Cubic Polynomial

For the cubic polynomial Q(x), it must give a remainder of 4 when divided by x+2. According to the Remainder Theorem, Q(-2) = 4. Let Q(x) = ax^3 + bx^2 + cx + d. Substitute x = -2 into Q(x) to get: \( a(-2)^3 + b(-2)^2 + c(-2) + d = 4 \). This simplifies to: \( -8a + 4b - 2c + d = 4 \). Again, we assume simplest form: Let a = 1, b = 2, c = 1 and solve for d: \( -8(1) + 4(2) - 2(1) + d = 4 \), resulting in d = 6. So, Q(x) = x^3 + 2x^2 + x + 6.
04

- Formulate the Quartic Polynomial

For the quartic polynomial R(x), it must give a remainder of 1 when divided by 2x-1. According to the Remainder Theorem, R(1/2) = 1. Let R(x) = ax^4 + bx^3 + cx^2 + dx + e. Substitute x = 1/2 into R(x) to get: \( a(1/2)^4 + b(1/2)^3 + c(1/2)^2 + d(1/2) + e = 1 \). This simplifies to: \( a/16 + b/8 + c/4 + d/2 + e = 1 \). We assume simplest form: Let a = 1, b = 2, c = 3, d = 4 and solve for e: \( 1/16 + 2/8 + 3/4 + 4/2 + e = 1 \), resulting in e = -34/16 = -17/8. So, R(x) = x^4 + 2x^3 + 3x^2 + 4x - 17/8.

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Key Concepts

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

Remainder Theorem
The Remainder Theorem is a fundamental concept in polynomial algebra. It states that for any polynomial function \(f(x)\), if you divide it by \(x - c\), the remainder of this division is simply \(f(c)\). This makes checking whether a specific value is a root of a polynomial much simpler. The theorem can also be paraphrased for a polynomial divided by \(ax - b\) to state that the remainder is \(f\left(\frac{b}{a}\right)\). This is incredibly useful when working with polynomials because it allows us to find remainders without needing to divide the entire polynomial.
Quadratic Polynomial
A quadratic polynomial is a polynomial of degree 2. Its general form is \(ax^2 + bx + c\). Quadratics often appear in various mathematical contexts, including equations of motion and area calculations. In this exercise, we needed to find a quadratic polynomial that gives a remainder of -4 when divided by \(x - 3\). According to the Remainder Theorem, this means \(P(3) = -4\). This helped us set up an equation to solve for the coefficients of the quadratic polynomial, ultimately finding \(P(x) = x^2 + x - 16\).
Cubic Polynomial
A cubic polynomial is a polynomial of degree 3, which means its highest power of \(x\) is \(x^3\). The general form is \(ax^3 + bx^2 + cx + d\). Cubic polynomials can describe more complex curves and have more varied applications, such as modelling population growth or certain types of economic scenarios. For this problem, we identified a cubic polynomial that leaves a remainder of 4 when divided by \(x + 2\). Using the Remainder Theorem, this translates to \(Q(-2) = 4\). By substituting -2 into the general form and solving, we concluded that \(Q(x) = x^3 + 2x^2 + x + 6\).
Quartic Polynomial
A quartic polynomial is a polynomial of degree 4, with the general form \(ax^4 + bx^3 + cx^2 + dx + e\). These polynomials are used in more advanced mathematical contexts, such as certain types of engineering problems or financial models. In this exercise, the quartic polynomial we needed to find had to leave a remainder of 1 when divided by \(2x - 1\), which according to the Remainder Theorem translates to \(R\left(\frac{1}{2}\right) = 1\). We used this condition to create an equation, eventually determining that \(R(x) = x^4 + 2x^3 + 3x^2 + 4x - \frac{17}{8}\).

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Most popular questions from this chapter

Explain how to determine the remainder when \(10 x^{4}-11 x^{3}-8 x^{2}+7 x+9\) is divided by \(2 x-3\) using synthetic division.

A design team determines that a cost-efficient way of manufacturing cylindrical containers for their products is to have the volume, \(V\) in cubic centimetres, modelled by \(V(x)=9 \pi x^{3}+51 \pi x^{2}+88 \pi x+48 \pi,\) where \(x\) is an integer such that \(2 \leq x \leq 8 .\) The height, \(h,\) in centimetres, of each cylinder is a linear function given by \(h(x)=x+3\) a) Determine the quotient \(\frac{V(x)}{h(x)}\) and interpret this result. b) Use your answer in part a) to express the volume of a container in the form \(\pi r^{2} h\) c) What are the possible dimensions of the containers for the given values of \(x ?\)

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.

Determine the equation with least degree for each polynomial function. Sketch a graph of each. a) a cubic function with zeros -3 (multiplicity 2 ) and 2 and \(y\) -intercept -18 b) a quintic function with zeros -1 (multiplicity 3 ) and 2 (multiplicity 2) and \(y\) -intercept 4 c) a quartic function with a negative leading coefficient, zeros -2 (multiplicity 2) and 3 (multiplicity 2), and a constant term of -6

The competition swimming pool at Saanich Commonwealth Place is in the shape of a rectangular prism and has a volume of \(2100 \mathrm{m}^{3} .\) The dimensions of the pool are \(x\) metres deep by \(25 x\) metres long by \(10 x+1\) metres wide. What are the actual dimensions of the pool?
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