1.02j Manipulate polynomials: expanding, factorising, division, factor theorem

It is given that \((x + 1)\) and \((x - 3)\) are two factors of \(f(x)\), where $$f(x) = px^3 - 3x^2 - 8x + q$$

1. Find the values of \(p\) and \(q\). [3 marks]
2. Fully factorise \(f(x)\). [2 marks]

Express \(3x^3 + 5x^2 - 27x + 10\) in the form \((x - 2)(ax^2 + bx + c)\), where \(a\), \(b\) and \(c\) are integers. [3 marks]

A student is looking for factors of the polynomial \(f(x)\) They suggest that \((x - 2)\) is a factor of \(f(x)\) The method they use to check this suggestion is to calculate \(f(-2)\) They correctly calculate that \(f(-2) = 0\) They conclude that their suggestion is correct.

1. Make one comment about the student's method. [1 mark]
2. Make two comments about the student's conclusion. [2 marks] 1 2

\(p(x) = x^3 - 5x^2 + 3x + a\), where \(a\) is a constant. Given that \(x - 3\) is a factor of \(p(x)\), find the value of \(a\) Circle your answer. [1 mark] \(-9\) \quad\quad \(-3\) \quad\quad \(3\) \quad\quad \(9\)

Find the coefficient of \(x\) in the expansion of $$(4x^3 - 5x^2 + 3x - 2)(x^5 + 4x + 1)$$ Circle your answer. $$-5 \quad -2 \quad 7 \quad 11$$ [1 mark]

\(p(x) = 2x^3 + 7x^2 + 2x - 3\)

1. Use the factor theorem to prove that \(x + 3\) is a factor of \(p(x)\) [2 marks]
2. Simplify the expression \(\frac{2x^3 + 7x^2 + 2x - 3}{4x^2 - 1}\), \(x \neq \pm \frac{1}{2}\) [4 marks]

\(x^2 + bx + c\) and \(x^2 + dx + e\) have a common factor \((x + 2)\) Show that \(2(d - b) = e - c\) Fully justify your answer. [4 marks]

\(p(x) = 4x^3 - 15x^2 - 48x - 36\)

1. Use the factor theorem to prove that \(x - 6\) is a factor of \(p(x)\). [2 marks]
2. 1. Prove that the graph of \(y = p(x)\) intersects the \(x\)-axis at exactly one point. [4 marks]
   2. State the coordinates of this point of intersection. [1 mark]

$$f(x) = 4x^3 - 12x^2 + 2x - 6$$

1. Use the factor theorem to show that \((x - 3)\) is a factor of \(f(x)\). [2]
2. Hence show that \(3\) is the only real root of the equation \(f(x) = 0\) [4]

1. Factorise completely \(x^3 + 10x^2 + 25x\) [2]
2. Sketch the curve with equation $$y = x^3 + 10x^2 + 25x$$ showing the coordinates of the points at which the curve cuts or touches the \(x\)-axis. [2]

The point with coordinates \((-3, 0)\) lies on the curve with equation $$y = (x + a)^3 + 10(x + a)^2 + 25(x + a)$$ where \(a\) is a constant.

1. Find the two possible values of \(a\). [3]

\(f(x) = -2x^3 - x^2 + 4x + 3\)

1. Use the factor theorem to show that \((3 - 2x)\) is a factor of \(f(x)\). [2]
2. Hence show that \(f(x)\) can be written in the form \(f(x) = (3 - 2x)(x + a)^2\) where \(a\) is an integer to be found. [4]

\includegraphics{figure_3} Figure 3 shows a sketch of part of the curve with equation \(y = f(x)\).

1. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
   1. \(f(x) \leq 0\)
   2. \(f'(\frac{x}{2}) = 0\)
   [3]

The function f is defined by $$f(x) = 4x^3 - 8x^2 - 51x - 45 \quad (x \in \mathbb{R})$$

1. 1. Fully factorise \(f(x)\) [2 marks]
   2. Hence, solve the inequality \(f(x) < 0\) [2 marks]
2. The graph of \(y = f(x)\) is translated by the vector \(\begin{pmatrix} 7 \\ 0 \end{pmatrix}\) The new graph is then reflected in the \(x\)-axis, to give the graph of \(y = g(x)\) Solve the inequality \(g(x) \leq 0\) [3 marks]

Use an algebraic method to solve the equation \(12x^3 - 29x^2 + 7x + 6 = 0\). Show all your working. [6]

The cubic polynomial \(f(x)\) is given by \(f(x) = 2x^3 + ax^2 + bx + c\), where \(a\), \(b\), \(c\) are constants. The graph of \(f(x)\) intersects the \(x\)-axis at the points with coordinates \((-3, 0)\), \((2.5, 0)\) and \((4, 0)\). Find the coordinates of the point where the graph of \(f(x)\) intersects the \(y\)-axis. [5]

Show that, for any value of the real constant \(b\), the equation \(x^3 - (b + 1)x + b = 0\) has \(x = 1\) as a solution. Find all values of \(b\) for which this equation has exactly two real solutions [5]

$$g(x) = 2x^3 + x^2 - 41x - 70$$

1. Use the factor theorem to show that \(g(x)\) is divisible by \((x - 5)\). [2]
2. Hence, showing all your working, write \(g(x)\) as a product of three linear factors. [4]

The finite region \(R\) is bounded by the curve with equation \(y = g(x)\) and the \(x\)-axis, and lies below the \(x\)-axis.

1. Find, using algebraic integration, the exact value of the area of \(R\). [4]

A curve has equation \(y = g(x)\). Given that • \(g(x)\) is a cubic expression in which the coefficient of \(x^3\) is equal to the coefficient of \(x\) • the curve with equation \(y = g(x)\) passes through the origin • the curve with equation \(y = g(x)\) has a stationary point at \((2, 9)\)

1. find \(g(x)\), [7]
2. prove that the stationary point at \((2, 9)\) is a maximum. [2]

The function \(f(x)\) is such that \(f(x) = -x^3 + 2x^2 + kx - 10\) The graph of \(y = f(x)\) crosses the \(x\)-axis at the points with coordinates \((a, 0)\), \((2, 0)\) and \((b, 0)\) where \(a < b\)

1. Show that \(k = 5\) [1 mark]
2. Find the exact value of \(a\) and the exact value of \(b\) [3 marks]
3. The functions \(g(x)\) and \(h(x)\) are such that $$g(x) = x^3 + 2x^2 - 5x - 10$$ $$h(x) = -8x^3 + 8x^2 + 10x - 10$$
   1. Explain how the graph of \(y = f(x)\) can be transformed into the graph of \(y = g(x)\) Fully justify your answer. [2 marks]
   2. Explain how the graph of \(y = f(x)\) can be transformed into the graph of \(y = h(x)\) Fully justify your answer. [2 marks]

In this question you should show all stages of your working. Solutions relying on calculator technology are not acceptable.

1. Using algebra, find all solutions of the equation $$3x^3 - 17x^2 - 6x = 0$$ [3]
2. Hence find all real solutions of $$3(y - 2)^6 - 17(y - 2)^4 - 6(y - 2)^2 = 0$$ [3]