Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number. When $P(x)$ is divided by $x - 3$ the remainder is 12. Find the remainder when $P(x)$ is divided by $x + 1$. --- The function $f(x) = ext{cos}(2x) - x$ has a zero near $x = rac{1}{2}$. Use one application of Newton's method to obtain another approximation to this zero. Give your answer correct to two decimal places. --- Find an expression for the coefficient of $x^2$ in the expansion of $\left(3x - 4 - \frac{8}{x} \right)^{8}$. --- Sketch the graph of the function $f(x) = 2\cos^{-1}(x)$. Clearly indicate the domain and range of the function. --- Alex's playlist consists of 40 different songs that can be arranged in any order. (i) How many arrangements are there for the 40 songs? (ii) Alex decides that she wants to play her three favourite songs first, in any order. How many arrangements of the 40 songs are now possible?

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Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number - HSC - SSCE Mathematics Extension 1 - Question 2 - 2011 - Paper 1

Question 2

Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number. When $P(x)$ is divided by $x - 3$ the remainder is 12. Find the remainder when $P(x)$ ... show full transcript

Worked Solution & Example Answer:Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number - HSC - SSCE Mathematics Extension 1 - Question 2 - 2011 - Paper 1

Step 1

a) Find the value of $P(3)$ and express it in terms of $a$ and $b$.

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Answer

To find the remainder when $P(x)$ is divided by $x - 3$ , we can use the polynomial remainder theorem, which states that the remainder of the division of a polynomial $P(x)$ by $x-c$ is $P(c)$ . Since $P(3) = 12$ , we have:

$P(3) = 3^3 - a(3^2) + 3 + b = 12$

This simplifies to:

$27 - 9a + 3 + b = 12$

Thus:

$-9a + b = 12 - 30$ $-9a + b = -18$ $b = 9a - 18.$

Next, we want to find the remainder when $P(x)$ is divided by $x + 1$ . To do that, we evaluate $P(-1)$ :

$P(-1) = (-1)^3 - a(-1)^2 + (-1) + b$

This reduces to:

$P(-1) = -1 - a - 1 + (9a - 18)$

$P(-1) = 9a - a - 18 - 2 = 8a - 20.$

Thus, the remainder when $P(x)$ is divided by $x + 1$ is $8a - 20$ .

Step 2

b) Use Newton's method to approximate the zero.

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Answer

Newton's method uses the formula:

$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$

First, we need to calculate $f'(x)$ :

$f'(x) = -2\sin(2x) - 1.$

Let’s choose $x_0 = \frac{1}{2}$ :

$f(\frac{1}{2}) = \cos(1) - \frac{1}{2} \approx 0.5403 - 0.5 = 0.0403$

$f'(\frac{1}{2}) = -2\sin(1) - 1 \approx -2(0.8415) - 1 = -2.6830.$

Now we apply Newton’s method:

$x_1 = \frac{1}{2} - \frac{0.0403}{-2.6830} \approx 0.5203.$

Thus, the approximation to two decimal places is $0.52$ .

Step 3

c) Find an expression for the coefficient of $x^2$ in the expansion.

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Answer

We begin with the binomial expansion:

$\left(3x - 4 - \frac{8}{x} \right)^{8}$

Using the multinomial expansion, we identify terms that contribute to the coefficient of $x^2$ . Set up the general term:

$T = \frac{8!}{k_1!...k_n!}(3x)^{k_1}(-4)^{k_2}\left(-\frac{8}{x}\right)^{k_3}$

where $k_1 + k_2 + k_3 = 8$ . We require that the total power of $x$ equals 2, thus:

ightarrow k_3 = k_1 + 6.$$ **Setting coefficients of like terms leads to:** - For $k_1 = 2$, $k_2 = 6$, $k_3 = 0$: $$\text{Coefficient} = \frac{8!}{2!6!} (3^2)(-4)^6 = 28(9)(4096) = 1152.$$ Thus, the coefficient of $x^2$ is $1152$.

Step 4

d) Sketch the graph of $f(x) = 2\cos^{-1}(x)$ and indicate the domain and range.

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Answer

The function $f(x) = 2\cos^{-1}(x)$ has the following characteristics:

Domain: The domain of $\cos^{-1}(x)$ is $[-1, 1]$ , thus the domain of $f(x)$ is also $[-1, 1]$ .
Range: The range of $\cos^{-1}(x)$ is $[0, \pi]$ , so the range for $f(x)$ will be $[0, 2\pi]$ .

To sketch the graph:

Start at $f(-1) = 2\cos^{-1}(-1) = 2\pi$
Move to $f(0) = \pi$
End at $f(1) = 0$

The graph will start at $(-1, 2 rac{ ext{π}}{1})$ , decrease to $(0, π)$ , and then to $(1, 0)$ , reflecting these values accordingly.

Step 5

e(i) How many arrangements are there for the 40 songs?

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Answer

The total arrangements of 40 songs can be calculated using the factorial formula:

$40!$

Which equals to:

$40! \approx 8.159 imes 10^{47}.$

Step 6

e(ii) How many arrangements of the 40 songs are now possible?

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Answer

If Alex plays her 3 favorite songs first in any order, we calculate:

Arrangements of the 3 songs: $3! = 6$ .
Remaining songs: $37!$ .

Thus, the total arrangements: $3! \times 37! = 6 \times 37! \approx 6.056 imes 10^{43}.$

Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number - HSC - SSCE Mathematics Extension 1 - Question 2 - 2011 - Paper 1

Worked Solution & Example Answer:Let $P(x) = x^3 - ax^2 + x + b$ be a polynomial, where $a$ is a real number - HSC - SSCE Mathematics Extension 1 - Question 2 - 2011 - Paper 1

a) Find the value of $P(3)$ and express it in terms of $a$ and $b$.

b) Use Newton's method to approximate the zero.

c) Find an expression for the coefficient of $x^2$ in the expansion.

d) Sketch the graph of $f(x) = 2\cos^{-1}(x)$ and indicate the domain and range.

e(i) How many arrangements are there for the 40 songs?

e(ii) How many arrangements of the 40 songs are now possible?

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