A-Level Edexcel Maths Pure Geometric Sequences & Series: Let $f(x) = x^3 + 4x^2 + x

Let $f(x) = x^3 + 4x^2 + x - 6.$ (a) Use the factor theorem to show that $(x + 2)$ is a factor of $f(x)$ - Edexcel - A-Level Maths Pure - Question 10 - 2007 - Paper 2

Question 10

Let $f(x) = x^3 + 4x^2 + x - 6.$ (a) Use the factor theorem to show that $(x + 2)$ is a factor of $f(x)$. (b) Factorise $f(x)$ completely. (c) Write down all the ... show full transcript

Worked Solution & Example Answer:Let $f(x) = x^3 + 4x^2 + x - 6.$ (a) Use the factor theorem to show that $(x + 2)$ is a factor of $f(x)$ - Edexcel - A-Level Maths Pure - Question 10 - 2007 - Paper 2

Step 1

Use the factor theorem to show that $(x + 2)$ is a factor of $f(x)$

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Answer

To use the factor theorem, we evaluate $f(-2)$ :

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

Calculating each term:

$(-2)^3 = -8$
$4(-2)^2 = 4 \times 4 = 16$
$(-2) = -2$
$-6 = -6$

Now substituting these values back into the function:

$f(-2) = -8 + 16 - 2 - 6 = 0$

Since $f(-2) = 0$ , it follows that $(x + 2)$ is a factor of $f(x)$ .

Step 2

Factorise $f(x)$ completely

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Answer

From part (a), we know that $(x + 2)$ is a factor. We can factor $f(x)$ as:

$f(x) = (x + 2)(x^2 + 2x - 3)$

Next, we factor the quadratic $x^2 + 2x - 3$ . To do this, we look for two numbers that multiply to $-3$ and add up to $2$ . These numbers are $3$ and $-1$ . Therefore:

$x^2 + 2x - 3 = (x + 3)(x - 1)$

Putting it all together, we have:

$f(x) = (x + 2)(x + 3)(x - 1)$

Step 3

Write down all the solutions to the equation $x^3 + 4x^2 + x - 6 = 0$

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Answer

Setting the factored form equal to zero:

$(x + 2)(x + 3)(x - 1) = 0$

This gives us the solutions:

$x + 2 = 0 \Rightarrow x = -2$
$x + 3 = 0 \Rightarrow x = -3$
$x - 1 = 0 \Rightarrow x = 1$

Thus, the solutions are:

$x = -3, -2, 1$

Let $f(x) = x^3 + 4x^2 + x - 6.$ (a) Use the factor theorem to show that $(x + 2)$ is a factor of $f(x)$ - Edexcel - A-Level Maths Pure - Question 10 - 2007 - Paper 2

Worked Solution & Example Answer:Let $f(x) = x^3 + 4x^2 + x - 6.$ (a) Use the factor theorem to show that $(x + 2)$ is a factor of $f(x)$ - Edexcel - A-Level Maths Pure - Question 10 - 2007 - Paper 2

Use the factor theorem to show that $(x + 2)$ is a factor of $f(x)$

Factorise $f(x)$ completely

Write down all the solutions to the equation $x^3 + 4x^2 + x - 6 = 0$

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