f(x) = 3x³ - 5x² - 58x + 40
(a) Find the remainder when f(x) is divided by (x - 3) - Edexcel - A-Level Maths Pure - Question 5 - 2010 - Paper 3
Question 5
f(x) = 3x³ - 5x² - 58x + 40
(a) Find the remainder when f(x) is divided by (x - 3).
Given that (x - 5) is a factor of f(x),
(b) find all the solutions of f(x) = 0.
Worked Solution & Example Answer:f(x) = 3x³ - 5x² - 58x + 40
(a) Find the remainder when f(x) is divided by (x - 3) - Edexcel - A-Level Maths Pure - Question 5 - 2010 - Paper 3
Step 1
Find the remainder when f(x) is divided by (x - 3)
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Answer
To find the remainder when dividing a polynomial by a linear term, we can use the Remainder Theorem. According to this theorem, the remainder of the division of the polynomial f(x) by (x - c) is f(c).
Here, we have:
f(3)=3(3)3−5(3)2−58(3)+40
Calculating this:
Calculate 33=27, so 3(33)=81.
Calculate 5(32)=5(9)=45.
Calculate 58(3)=174.
Substituting these values:
f(3)=81−45−174+40
Simplifying further gives:
f(3)=81−45=3636−174=−138−138+40=−98.
Thus, the remainder when f(x) is divided by (x - 3) is -98.
Step 2
find all the solutions of f(x) = 0
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Answer
Since (x - 5) is a factor of f(x), we can use polynomial long division or synthetic division to divide f(x) by (x - 5) to find the other factors.
Perform the division:
f(x)=(x−5)(3x2+10x−8).
Next, we need to solve for the quadratic equation:
3x2+10x−8=0.
Using the quadratic formula:
x=2a−b±b2−4ac
where a=3, b=10, and c=−8.
Calculate the discriminant:
b2−4ac=102−4(3)(−8)=100+96=196.
Substitute into the quadratic formula:
x=2×3−10±196
Simplifying gives:
x=6−10±14.
