Show that (x+2) is a factor of f(x) = x3 - 2x2 - 5x + 6. Then, factorise f(x) completely.

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Here's the solution to question 3: a) To show that (x+2) is a factor of f(x), we use the Factor Theorem. If (x+2) is a factor, then f(−2) must be equal to 0. Step 1: Substitute x = −2 into f(x). f(x) = x³ − 2x² − 5x + 6; f(−2) = (−2)³ − 2(−2)² − 5(−2) + 6 Step 2: Calculate the value of f(−2). f(−2) = −8 − 2(4) + 10 + 6; f(−2) = −8 − 8 + 10 + 6; f(−2) = −16 + 16; f(−2) = 0; Since f(−2) = 0, (x+2) is indeed a factor of f(x). b) To factorise f(x) completely, we can use polynomial division or synthetic division since we know (x+2) is a factor. Step 1: Perform polynomial division of f(x) by (x+2). (x³ − 2x² − 5x + 6) ÷ (x+2) Using synthetic division with root −2: -2 | 1 -2 -5 6 | -2 8 -6 ------------------ 1 -4 3 0 The quotient is x² − 4x + 3. Step 2: Factorise the quadratic quotient. x² − 4x + 3 We need two numbers that multiply to 3 and add to −4. These numbers are −1 and −3. (x−1)(x−3) Step 3: Write the complete factorisation of f(x). f(x) = (x+2)(x−1)(x−3) The complete factorisation of f(x) is (x+2)(x−1)(x−3).

Point	x	y
x-intercept	−2	0
Maximum	−0.79	8.21
y-intercept	0	6
x-intercept	1	0
Minimum	2.12	−4.06
x-intercept	3	0

Show that (x+2) is a factor of f(x) = x3 − 2x2 − 5x + 6. Then, factorise f(x) completely.

Answer

Step-by-step solution

Part a· 2 steps

Substitute $x = -2$ into $f(x)$ .

Calculate the value of $f(-2)$ .

Part b· 3 steps(x+2)(x-1)(x-3)

Perform polynomial division of $f(x)$ by $(x+2)$ .

Factorise the quadratic quotient.

Write the complete factorisation of $f(x)$ .

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