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

Expand and simplify \((n + 2)^3 - n^3\). [3]

Answer part (i) of this question on the insert provided. The insert shows the graph of \(y = \frac{1}{x}\).

1. On the insert, on the same axes, plot the graph of \(y = x^2 - 5x + 5\) for \(0 \leq x \leq 5\). [4]
2. Show algebraically that the \(x\)-coordinates of the points of intersection of the curves \(y = \frac{1}{x}\) and \(y = x^2 - 5x + 5\) satisfy the equation \(x^3 - 5x^2 + 5x - 1 = 0\). [2]
3. Given that \(x = 1\) at one of the points of intersection of the curves, factorise \(x^3 - 5x^2 + 5x - 1\) into a linear and a quadratic factor. Show that only one of the three roots of \(x^3 - 5x^2 + 5x - 1 = 0\) is rational. [5]

Factorise and hence simplify \(\frac{3x^2 - 7x + 4}{x^2 - 1}\). [3]

Simplify \((m^2 + 1)^2 - (m^2 - 1)^2\), showing your method. Hence, given the right-angled triangle in Fig. 10, express \(p\) in terms of \(m\), simplifying your answer. [4] \includegraphics{figure_3}

\(f(x) = 6x^3 + px^2 + qx + 8\), where \(p\) and \(q\) are constants. Given that \(f(x)\) is exactly divisible by \((2x - 1)\), and also that when \(f(x)\) is divided by \((x - 1)\) the remainder is \(-7\),

1. find the value of \(p\) and the value of \(q\). [6]
2. Hence factorise \(f(x)\) completely. [3]

f(x) = px³ + 6x² + 12x + q. Given that the remainder when f(x) is divided by (x - 1) is equal to the remainder when f(x) is divided by (2x + 1),

1. find the value of p. [4]

Given also that q = 3, and p has the value found in part (a),

1. find the value of the remainder. [1]

Figure 2 \includegraphics{figure_2} Figure 2 shows part of the curve with equation $$y = x³ - 6x² + 9x.$$ The curve touches the x-axis at A and has a maximum turning point at B.

1. Show that the equation of the curve may be written as $$y = x(x - 3)²,$$ and hence write down the coordinates of A. [2]
2. Find the coordinates of B. [5]

The shaded region R is bounded by the curve and the x-axis.

1. Find the area of R. [5]

f(x) = x³ + ax² + bx - 10, where a and b are constants. When f(x) is divided by (x - 3), the remainder is 14. When f(x) is divided by (x + 1), the remainder is -18.

1. Find the value of a and the value of b. [5]
2. Show that (x - 2) is a factor of f(x). [2]

\(f(n) = n^3 + pn^2 + 11n + 9\), where \(p\) is a constant.

1. Given that f(n) has a remainder of 3 when it is divided by \((n + 2)\), prove that \(p = 6\). [2]
2. Show that f(n) can be written in the form \((n + 2)(n + q)(n + r) + 3\), where \(q\) and \(r\) are integers to be found. [3]
3. Hence show that f(n) is divisible by 3 for all positive integer values of \(n\). [2]

\includegraphics{figure_3} Figure 3 shows part of the curve \(C\) with equation $$y = \frac{3}{5}x^2 - \frac{1}{4}x^3.$$ The curve \(C\) touches the \(x\)-axis at the origin and passes through the point \(A(p, 0)\).

1. Show that \(p = 6\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The curve \(C\) has a maximum at the point \(P\).

1. Find the \(x\)-coordinate of \(P\). [2]

The shaded region \(R\), in Fig. 3, is bounded by \(C\) and the \(x\)-axis.

1. Find the area of \(R\). [4]

f(x) = ax³ + bx² - 7x + 14, where a and b are constants. Given that when f(x) is divided by (x - 1) the remainder is 9.

1. write down an equation connecting a and b. [2 marks] Given also that (x + 2) is a factor of f(x),
2. find the values of a and b. [4 marks]

\includegraphics{figure_2} Figure 2 shows part of the curve C with equation y = f(x), where $$f(x) = x^3 - 6x^2 + 5x.$$ The curve crosses the x-axis at the origin O and at the points A and B.

1. Factorise f(x) completely [3 marks]
2. Write down the x-coordinates of the points A and B. [1 marks]
3. Find the gradient of C at A. [3 marks] The region R is bounded by C and the line OA, and the region S is bounded by C and the line AB.
4. Use integration to find the area of the combined regions R and S, shown shaded in Fig. 2. [7 marks]

$$f(x) = x^3 - x^2 - 7x + c, \text{ where } c \text{ is a constant.}$$ Given that \(f(4) = 0\),

1. find the value of \(c\), [2]
2. factorise \(f(x)\) as the product of a linear factor and a quadratic factor. [3]
3. Hence show that, apart from \(x = 4\), there are no real values of \(x\) for which \(f(x) = 0\). [2]

\(f(x) = 4x^3 + 3x^2 - 2x - 6\). Find the remainder when \(f(x)\) is divided by \((2x + 1)\). [3]

\(f(x) = x^3 - 19x - 30\).

1. Show that \((x + 2)\) is a factor of \(f(x)\). [2]
2. Factorise \(f(x)\) completely. [4]

\(f(x) = ax^3 + bx^2 - 7x + 14\), where \(a\) and \(b\) are constants. Given that when \(f(x)\) is divided by \((x - 1)\) the remainder is 9,

1. write down an equation connecting \(a\) and \(b\). [2]

Given also that \((x + 2)\) is a factor of \(f(x)\),

1. find the values of \(a\) and \(b\). [4]

Find the remainder when \(f(x) = 4x^3 + 3x^2 - 2x - 6\) is divided by \((2x + 1)\). [3]

\(\text{f}(n) = n^3 + pn^2 + 11n + 9\), where \(p\) is a constant.

1. Given that f\((n)\) has a remainder of \(3\) when it is divided by \((n + 2)\), prove that \(p = 6\). [2]
2. Show that f\((n)\) can be written in the form \((n + 2)(n + q)(n + r) + 3\), where \(q\) and \(r\) are integers to be found. [3]
3. Hence show that f\((n)\) is divisible by \(3\) for all positive integer values of \(n\). [2]

The cubic polynomial \(f(x)\) is given by $$f(x) = x^3 + ax + b,$$ where \(a\) and \(b\) are constants. It is given that \((x + 1)\) is a factor of \(f(x)\) and that the remainder when \(f(x)\) is divided by \((x - 3)\) is 16.

1. Find the values of \(a\) and \(b\). [5]
2. Hence verify that \(f(2) = 0\), and factorise \(f(x)\) completely. [3]

The polynomial f(x) is defined by \(f(x) = x^3 - 9x^2 + 7x + 33\).

1. Find the remainder when f(x) is divided by \((x + 2)\). [2]
2. Show that \((x - 3)\) is a factor of f(x). [1]
3. Solve the equation f(x) = 0, giving each root in an exact form as simply as possible. [6]

The cubic polynomial \(x^3 + ax^2 + bx - 6\) is denoted by f\((x)\).

1. The remainder when f\((x)\) is divided by \((x - 2)\) is equal to the remainder when f\((x)\) is divided by \((x + 2)\). Show that \(b = -4\). [3]
2. Given also that \((x - 1)\) is a factor of f\((x)\), find the value of \(a\). [2]
3. With these values of \(a\) and \(b\), express f\((x)\) as a product of a linear factor and a quadratic factor. [3]
4. Hence determine the number of real roots of the equation f\((x) = 0\), explaining your reasoning. [3]