The remainder when f(x) is divided by (x − a) is f(a), and f(a) = 0 makes (x − a) a factor, so this Year 12 sheet opens with the remainder theorem and its factor consequence. The answers record the constant remainder and f(2) = 0.
Key Takeaways
The remainder when f(x) is divided by (x − a) is f(a) (Q1).
f(a) = 0 makes (x − a) a factor of f(x) (Q2).
Q6 records f(2) = 0 as a factor condition.
Section A: Multiple Choice Questions (1 Mark Each)
Choose the correct option for each question.
1.The remainder theorem states that the remainder when f(x) is divided by (x − a) is
(a) f(a)(b) f'(a)(c) 0(d) a
2.If f(a) = 0, then (x − a) is
(a) a factor of f(x)(b) a constant(c) a divisor of degree 2(d) a root of f'(x)
3.When a polynomial is divided by a linear factor, the remainder is
(a) a constant(b) a linear expression(c) a quadratic(d) a fraction
4.The factor theorem is a special case of
(a) the remainder theorem(b) the binomial theorem(c) the quadratic formula(d) logarithms
5.A cubic equation has at most
(a) three real roots(b) two real roots(c) one real root(d) six real roots
6.If (x − 2) is a factor of f(x), then
(a) f(2) = 0(b) f(0) = 2(c) f(2) = 2(d) f(−2) = 0
Section B: Short Answer Type Questions (2 Marks Each)
Show all steps clearly.
7.State the remainder theorem.
8.State the factor theorem.
9.How do you find the remainder when f(x) is divided by (x − a)?
10.What is the degree of a cubic polynomial?
11.How many roots can a cubic equation have?
12.If f(3) = 0, what can you say about (x − 3)?
13.What is the remainder when a polynomial is divided by (x − 1)?
14.Give one application of the factor theorem.
Section C: Numericals & Word Problems (3 Marks Each)
Apply the concepts to solve the problems. Show all working.
15.For f(x) = x² + 2x + 1, use the remainder theorem to find the remainder when f(x) is divided by (x − 1).
16.For f(x) = x³ − 4x, use f(2) to check whether (x − 2) is a factor.
17.Factorise x² − 5x + 6.
18.For f(x) = x³ − 1, use f(1) to check whether (x − 1) is a factor.
19.For f(x) = 2x² + 3x − 2, find the remainder when f(x) is divided by (x − 1).
20.Factorise x² − 9.
Answer Key
1. a) f(a)
2. a) a factor of f(x)
3. a) a constant
4. a) the remainder theorem
5. a) three real roots
6. a) f(2) = 0
7. Refer to solution guide.
8. Refer to solution guide.
9. Refer to solution guide.
10. Refer to solution guide.
11. Refer to solution guide.
12. Refer to solution guide.
13. Refer to solution guide.
14. Refer to solution guide.
15. Refer to solution guide.
16. Refer to solution guide.
17. Refer to solution guide.
18. Refer to solution guide.
19. Refer to solution guide.
20. Refer to solution guide.
End of Worksheet
People Also Ask
What is the remainder when f(x) is divided by (x − a)?
f(a), option (a) in this worksheet (Q1).
What does f(a) = 0 tell us?
(x − a) is a factor of f(x), option (a) in this worksheet (Q2).
What does f(2) = 0 imply?
That (x − 2) is a factor, option (a) in this worksheet (Q6).
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