A-Level Edexcel Maths Pure Trigonometric Functions: f(x) = 2x³ - 7x²

f(x) = 2x³ - 7x² - 10x + 24 (a) Use the factor theorem to show that (x + 2) is a factor of f(x) - Edexcel - A-Level Maths Pure - Question 6 - 2012 - Paper 3

Question 6

f(x) = 2x³ - 7x² - 10x + 24 (a) Use the factor theorem to show that (x + 2) is a factor of f(x). (b) Factorise f(x) completely.

Worked Solution & Example Answer:f(x) = 2x³ - 7x² - 10x + 24 (a) Use the factor theorem to show that (x + 2) is a factor of f(x) - Edexcel - A-Level Maths Pure - Question 6 - 2012 - Paper 3

Step 1

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

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Answer

To apply the factor theorem, we need to evaluate f(-2):

$f(-2) = 2(-2)^3 - 7(-2)^2 - 10(-2) + 24$

Calculating each term:

$2(-2)^3 = 2(-8) = -16$
$-7(-2)^2 = -7(4) = -28$
$-10(-2) = 20$
$24 = 24$

Now, substituting these values:

$f(-2) = -16 - 28 + 20 + 24$ $f(-2) = -16 - 28 + 44$ $f(-2) = 0$

Since f(-2) = 0, by the factor theorem, (x + 2) is indeed a factor of f(x).

Step 2

Factorise f(x) completely.

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Answer

Given that (x + 2) is a factor, we can begin by performing polynomial long division.

Starting with:

$f(x) = 2x^3 - 7x^2 - 10x + 24$

Dividing by (x + 2):

First term: $2x^3 \div x = 2x^2$
Now multiply: $(x + 2)(2x^2) = 2x^3 + 4x^2$
Subtracting gives:

$(-7x^2 - 4x^2) = -11x^2$

Bring down -10x: $-11x^2 - 10x$
Next term: $-11x \div x = -11$
Multiply: $(x + 2)(-11) = -11x - 22$
Subtracting gives:

$(-10x + 22)$

This simplifies to:

$24 - 22 = 2$

So, we have:

$f(x) = (x + 2)(2x^2 - 11x + 12)$

Next, we factor $2x^2 - 11x + 12$ :

$2x^2 - 11x + 12 = (2x - 3)(x - 4)$

Thus, the complete factorization of f(x) is:

$f(x) = (x + 2)(2x - 3)(x - 4)$

f(x) = 2x³ - 7x² - 10x + 24 (a) Use the factor theorem to show that (x + 2) is a factor of f(x) - Edexcel - A-Level Maths Pure - Question 6 - 2012 - Paper 3

Worked Solution & Example Answer:f(x) = 2x³ - 7x² - 10x + 24 (a) Use the factor theorem to show that (x + 2) is a factor of f(x) - Edexcel - A-Level Maths Pure - Question 6 - 2012 - Paper 3

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

Factorise f(x) completely.

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