Exercises

1. Divide the polynomial \( 2x^3 - 7x^2 - 4x + 36 \) by each of the following divisors. Express the polynomial in terms of the quotient, divisor, and remainder. State any restrictions on the variable.
   1. \( x - 1 \)
   2. \( x + 3 \)
   3. \( 2x - 3 \)
2. Express each division in terms of the quotient, divisor, and remainder.
   1. \( \dfrac{x^3 + x - 8x^2 + 37}{x + 4}, x \neq -4 \)
   2. \( \dfrac{10x^3 - 9x^2 - 8x + 11}{5x - 2}, x \neq \frac{2}{5} \)
   3. \( \dfrac{3x + 2 - x^3}{x - 2}, x \neq 2 \)
   4. \( \dfrac{x^4 - 2x^2 + 12}{x - 4}, x \neq 4 \)
   5. \( \dfrac{x^5 - 2x^4 - 7x^3 + 13x^2 + 2x - 18}{x^2 - 2x - 3}, x \neq -1, 3 \)
3. The polynomial \( 6x^3 - 5x^2 - 49x + 60 \) is divided by \( 2x - 5 \).
   1. Identify the restrictions on \( x \).
   2. Show that the remainder is zero.
   3. Express the polynomial dividend in terms of the divisor, quotient, and remainder.
   4. What conclusion can be drawn when the remainder is zero?
   5. Express the polynomial in fully factored form.
4. 1. Divide \( f(x) = x^3 + (a + b)x^2 + (ab + c)x + ac \) by \( d(x) = x + a \) and express \( f(x) \) in terms of the divisor, quotient, and remainder.
   2. Using your findings in part a, create a cubic polynomial that has \( x - 2 \) as a factor. Verify your answer by carrying out the division.
5. 1. When dividing a \( 5^{\text{th}} \) degree polynomial by a \( 2^{\text{nd}} \) degree polynomial, what is the degree of the quotient and the maximum degree of the remainder?
   2. When dividing a \( n^{\text{th}} \) degree polynomial by a divisor of degree \( m \), where \(m\) and \(n\) are positive integers and \(m\leq n\), what is the degree of the quotient and the maximum degree of the remainder?
6. When a polynomial \( P(x) \) is divided by \( x + 3 \), the quotient is \( 3x^2 - 5x + 4 \) and the remainder is \( -10 \). Find \( P(x) \) in standard form.
7. Find the divisor given the divident, quotient, and remainder.
   1. The dividend is \( 3x^3 - 5x^2 - 7x - 1 \), the quotient is \( 3x^2 + 4x + 5 \), and the remainder is 14.
   2. The dividend is \( 2x^4 + 11x^3 + 5x^2 - 31x + 7 \), the quotient is \( 2x^2 + 3x - 5 \), and the remainder is \( -8x + 2 \).
8. The volume of a cylinder is given by \( (\pi x^3 + 4\pi x^2 - 3\pi x - 18 \pi) \text{ cm}^3 \). If the radius of the cylinder is \( (x + 3) \text{ cm} \), determine the height of the cylinder in terms of \( x \).
9. Given that
   \( \dfrac{x^4 - 3x^3 + px^2 - 11x - 7}{x^2 + 2x + 1} = x^2 + qx - 3 + \dfrac{r}{x^2 + 2x + 1} \) where \( p, q, r \in \mathbb{R} \),
   find \(p, q \), and \( r \).
10. 1. When a number \( n \) is divided by \( 7 \), the remainder is \( 3 \). What is the remainder when \( 4n \) is divided by \( 7 \)?
    2. When \( P(x) \) is divided by \( (x + 1) \), the remainder is \( 3 \). What is the remainder when \( xP(x) \) is divided by \( (x + 1) \)?
11. Prove that \( \underbrace{2^0 + 2^1 + 2^2 + 2^3 + \cdots + 2^{n - 2} + 2^{n - 1}}_{n \text{ terms}} = 2^n - 1 \), by dividing \( x^n - 1 \) by \( x - 1 \).

1. 1. Without dividing, determine the remainder when \( x^3 - 4x^2 + 7x - 6 \) is divided by
      1. \( x - 1 \)
      2. \( x + 3 \)
      3. \( x - 2 \)
      4. \( 2x - 1 \)
   2. Are any of the linear divisors in part a) factors of \( x^3 - 4x^2 + 7x - 6 \)?
2. 1. When \( x^3 + kx^2 - 4x + 2, k \in \mathbb{R}, \) is divided by \( x + 2 \), the remainder is \( 26 \). Find \( k \).
   2. The remainder is \( 4 \) when \( 2x^2 - 3x + 5 \) is divided by \( x + k, k \in \mathbb{R} \). Find \( k \).
3. 1. Without dividing, determine which of the following are factors of \( 2x^4 - 15x^3 + 11x^2 + 78x - 40 \)?
      1. \( x + 1 \)
      2. \( x + 2 \)
      3. \( x - 5 \)
      4. \( x + 4 \)
   2. Find the remaining factors of the polynomial \( 2x^4 - 15x^3 + 11x^2 + 78x - 40 \).
4. Determine the value(s) of \( k \), \( k \in \mathbb{R} \):
   1. if \( x - 5 \) is a factor of \( x^3 + 2x^2 + kx + 30 \)
   2. if \( 2x + 3 \) is a factor of \( 2x^3 + kx^2 - 2x + 15 \)
   3. if \( x + k \) is a factor of \( 2x^2 + kx - 9 \)
   4. if \( 2x + 1 \) and \( x - k \) are factors of \( nx^2 - 9x - n, n \in \mathbb{R} \)
5. State the equation of any cubic polynomial that has a remainder of \( -6 \) when divided by \( x + 3 \).
6. The polynomial \( 2x^3 + px^2 + qx + 12, p, q \in \mathbb{R}, \) has a factor of \( x + 3 \) and a remainder of \( -10 \) when divided by \( x - 2 \). Find \( p \) and \( q \).
7. For what value(s) of \( k \), \( k \in \mathbb{R} \), does the function \( f(x) = x^3 + 6x^2 + kx - 4 \) have the same remainder when divided by either \( x - 1 \) or \( x + 2 \)?
8. Find the value of \( a \) and \( b \) if \( x^2 - 5x + 4 \) is a factor of the polynomial \( 2x^3 + ax^2 + bx - 4 \). Express the polynomial in factored form.
9. Given that \(a\) and \(b\) are non-zero integers and \(n\) is a positive integer:
   1. for what values of \( n \) is \( (a - b) \) a factor of \( a^n - b^n \)
   2. for what values of \( n \) is \( (a + b) \) a factor of \( a^n + b^n \)
10. A polynomial \( P(x) \) has a remainder of \( 3 \) when divided by \( x - 2 \) and a remainder of \( -5 \) when it is divided by \( x + 2 \). Determine the remainder when the polynomial is divided by \( x^2 - 4 \).

1. Given the polynomial \( P(x) = 2x^3 + 5x^2 - 8x - 15 \)
   1. Using the rational root theorem, list the potential rational roots of \( P(x) = 0 \).
   2. Show \( P(-3) = 0 \). What is a linear factor of \( P(x) \)?
   3. Determine the corresponding quadratic factor.
2. Given the polynomial \( P(x) = 4x^3 + x^2 - 7x + 3 \)
   1. Using the rational root theorem, list the potential rational roots of \( P(x) = 0 \).
   2. Show \( P\left( \frac{3}{4} \right) = 0 \). What is a linear factor of \( P(x) \)?
   3. Determine the corresponding quadratic factor.
3. Given the polynomial \( P(x) = 4x^3 + 17x^2 + 18x + 9 \)
   1. List all potential rational roots of \( P(x) = 0 \).
   2. The roots of \( P(x) = 0 \) must be negative. Explain.
   3. Using the factor theorem, find a factor of \( P(x) \).
   4. Determine the corresponding quadratic factor using the “have and need” method.
4. Determine the value of \( k \) if \( x - 1 \) is a factor of the polynomial \( 14x^3 + kx^2 - 34x + 12 \). Express the polynomial in factored form.
5. Factor fully.
   1. \( x^3 - x^2 + x - 1 \)
   2. \( 2x^3 + 11x^2 + 12x - 9 \)
   3. \( x^3 - 7x - 6 \)
   4. \( 3x^3 - 3x^2 + 6x - 24 \)
   5. \( 5x^3 + 3x^2 - 12x + 4 \)
   6. \( x^3 + 9x^2 + 8x - 60 \)
   7. \( x^4 - 5x^2 + 4 \)
   8. \( x^4 + 3x^3 - 38x^2 + 24x + 64 \)
6. Show that \( x - y \) is a factor of the polynomial \( x^3 - 2x^2y - 5xy^2 + 6y^3 \). Factor this polynomial fully.
7. If \( x - 2 \) and \( x + 2 \) are factors of \( 6x^3 + ax^2 + bx + 16 \), determine the values of \( a \) and \( b \), and any remaining factors.
8. Determine the values of \( m \) and \( n \) if \( 3x^2 - x - 2 \) is a factor of the polynomial \( 3x^4 + mx^3 - 19x^2 + nx + 12 \). Express the polynomial in factored form.
9. The function \( f(x) = ax^3 + bx^2 + cx - 8 \) has zeros at \( x = -1 \) and \( x = 2 \), and passes through the point \( (1, -2) \). Find the remaining zero and sketch the graph of the function.
10. Suppose \( P(x) \) is a quadratic whose coefficients are all odd integers. Prove that \( P(x)=0 \) has no rational roots.

1. Factor the following polynomials using the factor theorem.
   1. \( x^3 - 4x^2 + x + 6 \)
   2. \( x^3 + 8x^2 + 21x + 18 \)
   3. \( x^4 - x^3 - 3x^2 + x + 2 \)
2. Factor by grouping.
   1. \( x^3 - 2x^2 - 2x + 4 \)
   2. \( -3x^3 + 6x^2 + 7x - 14 \)
   3. \( x^3 - 3x^2 - 4x + 12 \)
   4. \( 2x^3 + 10x^2 - 4x - 20 \)
   5. \( x^7 - 3x^6 + x^5 - 3x^4 + x^3 - 3x^2 + x - 3 \)
3. Factor each expression.
   1. \( x^3 - 8 \)
   2. \( 27x^3 + 1 \)
   3. \( 625x^3 - 40 \)
   4. \( 125 - 64x^3 \)
4. Factor the following trinomials.
   1. \( x^4 - 5x^2 + 4 \)
   2. \( x^4 - 2x^2 + 1 \)
   3. \( x^6 + 4x^3 + 4 \)
   4. \( 4y^4 + 11y^2 - 3 \)
   5. \( 2x^4 - 22x^2 + 36 \)
5. Factor fully, using an appropriate method.
   1. \( 5t^3 + 3t^2 - 12t + 4 \)
   2. \( x^4 - 5x^3 + 2x^2 + 8x \)
   3. \( 2y^3 - 6y^2 + y - 3 \)
   4. \( 3n^4 + 6n^2 + 3 \)
   5. \( 3x^4 + 2x^2 - 5 \)
   6. \( 3y^4 - 243 \)
   7. \( 2x^5 - 3x^4 - 16x^2 + 24x \)
   8. \( 6u^5 - 2u^4 - 9u^2 + 3u \)
   9. \( x^6 - 16x^3 + 64 \)
6. Factor fully.
   1. \( 5(1 - p)^2 + 20(1 - p) + 15 \)
   2. \( (2x + 3)^3 - (x + 1)^3 \)
   3. \( 4(x - 2)^3 - 7(x - 2) - 3 \)
   4. \( 2\sin^2{(x)} + 7\sin{(x)} + 3 \)
   5. \( \cos^3{(x)} - 1 \)
   6. \( \sin{(x)}\cos{(x)} - \cos{(x)} + 3\sin{(x)} - 3 \)
   7. \( 3^{2x} - 10(3^{x}) + 9 \)
   8. \( 2^{3x} - 1 \)
7. Factor fully.
   1. \( 3x^2 - 5xy - 2y^2 \)
   2. \( x^4y^4 - z^4 \)
   3. \( x^3 + 8y^3 \)
   4. \( x^4y^4 - 13x^2y^2 + 36 \)
   5. \( x^3 + x^2y - xy - y^2 \)
   6. \( (x + y)^3 - z^3 \)
8. 1. Given \( f(x) = 2x^4 + 3x^3 - 5x^2 + 3x + 2 \). If \(k\) is a non-zero real root of \(f(x)=0\), show that \(\frac{1}{k}\) is a also a root.
   2. Let \( p(x) = 0 \) be an equation whose roots are all non-zero real numbers. State a sufficient condition on the coefficients of \(p(x)\) that will ensure that the roots of \(p(x)=0\) occur in reciprocal pairs. That is, if \(k\) is a root of \(p(x),\ k\neq 0,\ k\in\mathbb{R}\), then \(\frac{1}{k}\) must also be a root. Prove your assertion for fifth and sixth degree polynomials.
   3. Factor the polynomial \( q(x) = 42x^4 - 299x^3 + 614x^2 - 299x + 42 \), given that \( P(3) = 0 \).
9. Factor fully.
   1. \( abx^3 + (a + b - ab)x^2 + (1 - a - b)x - 1 \)
   2. \( x^3 - (p + q - r)x^2 + (pq - pr - qr)x + pqr \)
10. The factors of \( 5x^2 + kxy - 6y^2 + 13x - 5y + 6 \) are of the form \( (ax + by + c)(dx + ey + f) \), where \( a, b, c, d, e, f \) are integers. Find the value of \( k \).
11. 1. Factor \( x^{12}-1 \) fully.
    2. List all polynomials of the form \(x^4 + bx^3 +cx^2 + dx + e\) with rational coefficients that are factors of the polynomial, \(x^{12}-1\).

1. Solve the following polynomial equations by factoring where \( x \in \mathbb{R} \).
   1. \( x^3 - 5x^2 - 4x + 20 = 0 \)
   2. \( 2x^3 + 3x^2 = 11x + 6 \)
   3. \( 4x^2 = x^3 + 2x + 3 \)
   4. \( x^4 - 7x^2 + 12 = 0 \)
   5. \( 2x^3 + 15 = 6x^2 + 5x \)
   6. \( 2x(x^3 + 1) = x^2(4x + 1) \)
   7. \( 2x^4 + 8x + 12 = 3x^2(x + 3) \)
2. Find all possible roots of the polynomial equation where \( x \in \mathbb{C} \).
   1. \( 2x^3 + 5x^2 + 14x + 6 =0 \)
   2. \( 8x^4 = x \)
   3. \( x^2(4x^2 + 17) = 15 \)
3. If one root in each of the given equations is \( x=2 \), determine the other roots. In the following equations, \( k \in \mathbb{R} \).
   1. \( 3x^3 - 15x^2 + kx - 4 = 0 \)
   2. \( 25x^4 + kx^2 + 16 = 0 \)
4. Sketch a possible graph for each polynomial function, using the intercepts and end behaviour of the function.
   1. \( y = 2x^3 - 12x^2 + 18x \)
   2. \( y = -x^3 + 4x^2 + x - 4 \)
   3. \( y = x^4 - 8x^2 + 16 \)
5. Explain why
   1. \( 15x^5 + 4x^4 + 9x^2 + 7x + 380 = 0 \) has at least one real root.
   2. \( 5x^6 + 3x^4 + 8x^2 + 120 = 0 \) has no real roots.
6. A rectangular holding tank is \( x \) metres deep, \( (6x - 8) \) metres long, and \( (6x - 16) \) metres wide. Find the dimensions of the tank with a volume of \( 512 \text{ m}^3 \).
7. The product of the squares of two consecutive integers is \( 1764 \). Find all possible values for the integers.

8. A rectangular sheet of metal with dimensions \( 20 \) cm by \( 15 \) cm is to be used to create an open top box by cutting a square, \(x\ \text{cm}\) by \(x\ \text{cm}\), from each corner and bending up the sides. If a volume of \( 375~\text{cm}^3 \) is required, determine the side length of the squares that must be cut.

   

9. A box with a lid is to be created from a \( 50 \) cm by \( 30 \) cm piece of cardboard by cutting \( x \) by \( x \) squares from the four corners of the cardboard, and at the centre of the two sides, as shown in the diagram. Determine the function that represents the volume of the box in terms of \( x \), and state the restrictions on \( x \). If the box is to have a volume of \( 1750 \text{ cm}^3 \), determine the side length of the squares that need to be cut

   
10. Solve \( x^2(x^2 + 6) = 5x^3 - x + 1, x \in \mathbb{R} \).

11. The first three square pyramidal numbers are \( 1, 5, \) and \( 14 \) as shown in the diagram. The number of balls in each layer of the pyramids is a perfect square.

    

    The only pyramidal number, other than \( 1 \), that is a perfect square is \( 4900 \). How many layers are in the pyramid which contains 4900 balls?

    Note: The sum of the first \( n \) perfect squares is \( \frac{n(n + 1)(2n + 1)}{6} \).

12. Determine all possible values of \( n, n \in \mathbb{R} \), such that the equation \( 3x^3 + 11x^2 + 8x + n = 0 \) has two equal real roots.

1. Solve each of the following polynomial inequalities using a graphical approach, \( x \in \mathbb{R} \).
   1. \( -2x(x + 2)(x - 3) \lt 0 \)
   2. \( (x + 4)(x + 1)(x - 2)^2 \leq 0 \)
2. Solve each of the following polynomial inequalities using an interval sign table, \( x \in \mathbb{R} \).
   1. \( 2(x + 3)(x - 1)(x - 5) \leq 0 \)
   2. \( -3(x + 4)(x - 3)^3 \gt 0 \)
3. Solve the following polynomial inequalities, \( x \in \mathbb{R} \).
   1. \( x^2 - 4x + 3 \lt 0 \)
   2. \( x^3 - 3x - 2 \geq 0 \)
   3. \( x^4 - 1 \geq 0 \)
   4. \( -x^2 + 3x + 1 \lt 0 \)
   5. \( -2x^4 - 2x^3 + 16x^2 + 24x \lt 0 \)
4. Solve the following polynomial inequalities, \( x \in \mathbb{R} \).
   1. \( x^2 + x \gt 6 \)
   2. \( x^3 \lt x \)
   3. \( x^3 + 2 \geq 2x^2 + x \)
   4. \( x^3 \geq x^2 \)
   5. \( 2x^2 - 2x \geq 2 - x \)
   6. \( x^4 \lt 22x^2 + 75 \)

5. An object is thrown into the air at a speed of \( v_0 \) m/s from a height of \(h_0\) meters. The height \(h\), in metres, of the object after \(t\) seconds is given by the equation

   \[ h(t) = -4.9t^2 + v_0\sin(\theta) t + h_0 \]

   where \( \theta \) is the angle between the object and the horizontal.

   A \( 1.8 \) metre tall quarterback throws a ball at a speed of \( 5.6 \text{ m/s} \) to a receiver, at an angle of \( 30 \) degrees above horizontal. For how long is the ball above the quarterback's head? (You may assume that the ball was released at a height of \(1.8\) metres.)

6. Let \( f(x) = -2x + 1, g(x) = x^2 - 2x + 1 \) and \( h(x) = x^3 - 1 \). Determine all values of \( x \) such that\[ f(x) \lt g(x) \lt h(x) \] and illustrate the situation graphically.
7. The number \( n \) (in hundreds), of mosquitoes in a camping area after \( t \) weeks can be modelled by the equation\[ n(t) = 2t^4 - 5t^3 - 16t^2 + 45t \] According to this model, when will the population of mosquitoes be greater than \( 1800 \)?
8. A zoo wishes to construct an aquarium in the shape of a rectangular prism such that the length is twice the width and \(5\ \text{m}\) greater than the height. If the aquarium must have a volume strictly between \( 1125 \text{ m}^3 \) and \( 3000 \text{ m}^3 \), determine the restrictions on the length of the aquarium.

9. Determine the equation of a quintic function \( f(x) \) that satisfies the following conditions:

   • \( f(-3) = f(0) = f(4) = 0 \)
   • \( f(1) = -9 \)
   • \( f(x) \gt 0 \) when \( x \lt -3 \) or \( -3 \lt x \lt 0 \)
   • \( f(x) \lt 0 \) when \( 0 \lt x \lt 4 \) or \( x \gt 4 \)

   Illustrate the situation graphically.

10. The solution to \( x^2 + bx + 24 \lt 0 \) is the set of all values of \( x \) such that \( k \lt x \lt k + 2 \) for some real value of \( k \). Determine all possible values of \( b,\ b\in\mathbb{R} \). Justify your answer.
11. A quartic function has turning points at \( (-3,0), (1, 0) \), and \( (-1, -16) \). Determine all values of \( x \) such that \( -9 \lt f(x) \lt 0 \).

Use technology, such as graphing software or a graphing calculator, to assist in solving the problems presented in this exercise. Whenever applicable, express answers to \( 2 \) decimal places of accuracy, unless stated otherwise.

1. Solve each polynomial equation or inequality using technology.
   1. \( 3x^3 - x^2 + 8x - 4 \lt 0 \)
   2. \( -x^3 - 3x \leq 9x^2 - 8 \)
   3. \( -x^4 + 7x^3 + 5 = x(10x + 7) \)
   4. \( -x^4 + 3x^3 + 9x^2 \gt 5x + 5 \)
2. A car manufacturer has developed a formula\[ b = 0.00343v^2 + 0.5365v \] to describe the minimum braking distance of a car, \(b\) in meters, travelling at \( v \) kilometres per hour. For which speeds will the car be able to stop within a distance of \(100\) metres?
3. The population of a certain city in Northern Ontario can be modelled by\[ P(t) = 65 - 0.0025t^3 + 3.32t \] where the population, \( P \), is in thousands and the time, \( t \), is in years from \( 2014 \).
   1. Graph the function using technology.
   2. What will be the population of the city in \(2020 \)?
   3. In what year will the population reach \( 75\,000 \)?
   4. During what period of time will the population of the city be in the range \( 80\,000 \) to \( 100\,000 \) people?
4. The height \( h \) (in metres) of a certain section of a roller coaster can be modelled by the equation\[ h(t) = -0.003t^4 + 0.156t^3 - 2.535t^2 + 12.75t + 11.7,\ 0\leq t \leq 25 \] where \( t \) is the time in seconds elapsed after beginning the ride.
   1. Graph the function using technology.
   2. While travelling this section of the ride, when will the rider be
      1. 20 metres above ground?
      2. at least 10 metres above ground?
      3. less than 7 metres above ground?
   3. Estimate the maximum and minimum heights of the ride in this section, and when the rider reaches these heights. Answer to \( 1 \) decimal place of accuracy.
5. The annual revenue of a small family restaurant is modeled by the function\[ R(t) = 0.8t^3 - 8.5t^2 + 28t, t \geq 0 \] where \( R \) is the revenue, in tens of thousands, and \( t \) is the number of years since the restaurant was opened in \( 2009 \).
   1. How much revenue did the restaurant make in its first year?
   2. According to this model, in which year(s) will the annual revenue of the company be \( $ 800\,000 \)?
   3. The annual cost \( C \) (in tens of thousands) to maintain the restaurant is given by\[ C(t) = 0.82t + 30.5 \] During which year(s) was the restaurant making a profit?
   4. The owners plan to purchase a larger property when their annual profit exceeds \( 4.5 \) million. Based on this model, when will this purchase be made?
6. A company that produces MP3 players estimates that the profit, \( P \) (in tens of thousands of dollars), for selling their top model is given by\[ P(a) = -0.075a^3 + 4.62a^2 - 340 \] where \( a \) is the dollar amount spent on advertising (in tens of thousands).
   1. How much must be spent on advertising, to the nearest thousand, to ensure a profit of at least \( 5 \) million?
   2. According to this model, estimate the ideal amount the company should spend on advertisement, to the nearest thousand.
7. The height of a cylinder is \( 5 \text{ cm} \) longer than the radius.
   1. Find the dimensions of the cylinder with volume \( 375 \text{ cm}^3 \).
   2. What restrictions must be placed on the radius for the volume of the cylinder to be greater than \( 250 \text{ cm}^3 \) but less than \( 500 \text{ cm}^3 \)?