A-Level Edexcel Maths Pure Trigonometric Equations: (a) Use the factor theorem to sh

(a) Use the factor theorem to show that $(x+4)$ is a factor of $2x^3 - 25x + 12$ - Edexcel - A-Level Maths Pure - Question 5 - 2005 - Paper 2

Question 5

(a) Use the factor theorem to show that $(x+4)$ is a factor of $2x^3 - 25x + 12$. (b) Factorise $2x^3 - 25x + 12$ completely.

Worked Solution & Example Answer:(a) Use the factor theorem to show that $(x+4)$ is a factor of $2x^3 - 25x + 12$ - Edexcel - A-Level Maths Pure - Question 5 - 2005 - Paper 2

Step 1

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Answer

To apply the factor theorem, we need to evaluate the polynomial at the root of the factor, which in this case is $x = -4$ .

First, calculate:

$f(-4) = 2(-4)^3 - 25(-4) + 12$

Calculating term by term:

eq 0$$

Since $f(-4) \neq 0$ , $(x + 4)$ is not a factor of $2x^3 - 25x + 12$ .

Step 2

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Answer

To factor the polynomial $2x^3 - 25x + 12$ , we can initially try synthetic division or polynomial long division with possible rational roots.

Assuming possible roots, after confirming they do not work, we can rewrite the polynomial as follows:

Group similar terms and factor out the common coefficients:

$2x^3 - 25x + 12 = (x - 3)(2x^2 + 6x - 4)$

Next, we need to factor $2x^2 + 6x - 4$ . We can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 2$ , $b = 6$ , and $c = -4$ :

So then, we factor further:

Once the roots are calculated, we will have the complete factorization for $2x^3 - 25x + 12$ as:

$2x^3 - 25x + 12 = (x - 3)(2x + 4)(x - 1)$