f(x) = 2x^3 - 3x^2 - 39x + 20
(a) Use the factor theorem to show that (x + 4) is a factor of f(x) - Edexcel - A-Level Maths Pure - Question 2 - 2008 - Paper 2
Question 2
f(x) = 2x^3 - 3x^2 - 39x + 20
(a) Use the factor theorem to show that (x + 4) is a factor of f(x).
(b) Factorise f(x) completely.
Worked Solution & Example Answer:f(x) = 2x^3 - 3x^2 - 39x + 20
(a) Use the factor theorem to show that (x + 4) is a factor of f(x) - Edexcel - A-Level Maths Pure - Question 2 - 2008 - Paper 2
Step 1
Use the factor theorem to show that (x + 4) is a factor of f(x).
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Answer
To use the factor theorem, we substitute x = -4 into f(x):
f(−4)=2(−4)3−3(−4)2−39(−4)+20
Calculating the values:
The first term: 2(−4)3=2(−64)=−128.
The second term: −3(−4)2=−3(16)=−48.
The third term: −39(−4)=156.
Finally, the constant term: 20.
Combining these results:
f(−4)=−128−48+156+20=0
Since f(−4)=0, it follows that (x+4) is a factor of f(x).
Step 2
Factorise f(x) completely.
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Answer
To factorize f(x) completely, we can start by using polynomial long division or synthetic division to divide f(x) by (x+4):
Performing the division:
f(x)=(x+4)(2x2−11x+5)
Next, we need to factor the quadratic 2x2−11x+5:
Using the quadratic formula:
x=2a−b±b2−4ac
where a=2, b=−11, and c=5:
x=2(2)11±(−11)2−4(2)(5)x=411±121−40x=411±81x=411±9
This gives two roots:
x=420=5
and
x=42=21
Thus, the complete factorization of f(x) is:
f(x)=(x+4)(2x−1)(x−5)
