f(x) = 3x^3 - 5x^2 - 16x + 12.
(a) Find the remainder when f(x) is divided by (x - 2).
Given that (x + 2) is a factor of f(x),
(b) factorise f(x) completely.
Worked Solution & Example Answer:f(x) = 3x^3 - 5x^2 - 16x + 12 - Edexcel - A-Level Maths Pure - Question 4 - 2007 - Paper 2
Step 1
Find the remainder when f(x) is divided by (x - 2)
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Answer
To find the remainder, we can use the Remainder Theorem which states that the remainder of the division of a polynomial f(x) by (x - c) is equal to f(c).
Here, c = 2.
Calculate f(2):
f(2)=3(2)3−5(2)2−16(2)+12
Calculating each term:
3(2)3=3(8)=24
−5(2)2=−5(4)=−20
−16(2)=−32
+12=12
Now combine these:
f(2)=24−20−32+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
To factorise f(x), we know (x + 2) is a factor. We will use polynomial long division to divide f(x) by (x + 2).
Performing the division:
Divide the leading terms: 3x3÷x=3x2.
Multiply (x + 2) by 3x^2: 3x2(x+2)=3x3+6x2.
Subtract this result from f(x):
f(x)−(3x3+6x2)=−5x2−6x2−16x+12=−11x2−16x+12
Now repeat the process: Divide the leading terms: −11x2÷x=−11x.
Multiply (x + 2) by -11x: −11x(x+2)=−11x2−22x.
Subtract again:
−11x2−16x+12−(−11x2−22x)=6x+12
Finally, divide 6x + 12 by (x + 2): It gives us 6 as the remaining constant term.
Putting it all together:
f(x)=(x+2)(3x2−11x+6)
Now we factor the quadratic: 3x2−11x+6.
Using factoring we find:
3x2−11x+6=(x−2)(3x−3)
So, the complete factorization of f(x) is:
f(x)=(x+2)(x−2)(3x−3)
Or simplifying: f(x)=(x+2)(x−2)(3(x−1)).
