A-Level Edexcel Maths Pure Integration: f(x) = 3x^3 - 5x^2

f(x) = 3x^3 - 5x^2 - 16x + 12 - Edexcel - A-Level Maths Pure - Question 5 - 2007 - Paper 2

Question 5

f(x) = 3x^3 - 5x^2 - 16x + 12. (a) Find the remainder when divided by (x - 2). (b) Given that (x + 2) is a factor of f(x), factorise f(x) completely.

Worked Solution & Example Answer:f(x) = 3x^3 - 5x^2 - 16x + 12 - Edexcel - A-Level Maths Pure - Question 5 - 2007 - Paper 2

Step 1

Find the remainder when divided by (x - 2)

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Answer

To find the remainder when dividing a polynomial by (x - 2), we can use the Remainder Theorem. This theorem states that the remainder of the polynomial f(x) when divided by (x - c) is equal to f(c). Here, c = 2.

Calculating f(2):

f(2) = 3(2)^3 - 5(2)^2 - 16(2) + 12 = 3(8) - 5(4) - 32 + 12 = 24 - 20 - 32 + 12 = -16.

Thus, the remainder when f(x) is divided by (x - 2) is -16.

Step 2

factorise f(x) completely

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Answer

Since (x + 2) is a factor of f(x), we can use polynomial long division to find the complete factorization of f(x).

We divide f(x) by (x + 2) as follows:

Dividing the first term: $rac{3x^3}{x} = 3x^2$
Multiply (x + 2) by 3x^2: $3x^2(x + 2) = 3x^3 + 6x^2$
Subtract from f(x): $o (3x^3 - 5x^2 - 16x + 12) - (3x^3 + 6x^2) = -11x^2 - 16x + 12$
Repeat this process: $rac{-11x^2}{x} = -11x$ Multiply: $-11x(x + 2) = -11x^2 - 22x$ Subtract again: $(-11x^2 - 16x + 12) - (-11x^2 - 22x) = 6x + 12$
Finally: $rac{6x}{x} = 6$ Multiply: $6(x + 2) = 6x + 12$ Subtract: $6x + 12 - (6x + 12) = 0$

Thus, we have: $f(x) = (x + 2)(3x^2 - 11x + 6).$

Next, we factor this quadratic: 3x^2 - 11x + 6. To factor it, we need two numbers that multiply to (3 * 6 = 18) and add to -11. These numbers are -9 and -2.

Thus, we can factor as: $3x^2 - 9x - 2x + 6 = 3x(x - 3) - 2(x - 3) = (3x - 2)(x - 3).$

Combining this, $f(x) = (x + 2)(3x - 2)(x - 3).$

So the complete factorization of f(x) is (x + 2)(3x - 2)(x - 3).

f(x) = 3x^3 - 5x^2 - 16x + 12 - Edexcel - A-Level Maths Pure - Question 5 - 2007 - Paper 2

Worked Solution & Example Answer:f(x) = 3x^3 - 5x^2 - 16x + 12 - Edexcel - A-Level Maths Pure - Question 5 - 2007 - Paper 2

Find the remainder when divided by (x - 2)

factorise f(x) completely

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