Polynomials

A polynomial is an expression with multiple terms involving powers of a variable, like x³ − 2x² + x − 5. The remainder theorem states that dividing a polynomial f(x) by (x − a) leaves a remainder equal to f(a); the factor theorem is a special case — if f(a) = 0, then (x − a) is a factor of f(x). These theorems let you test for and find factors of a polynomial without doing full polynomial long division every time.

Key terms

Polynomial:

An expression with multiple terms involving powers of a variable.

Remainder theorem:

Dividing f(x) by (x − a) leaves a remainder of f(a).

Factor theorem:

If f(a) = 0, then (x − a) is a factor of f(x).

Questions

1. 1. A polynomial is:

   • An expression with multiple terms involving powers of a variable
   • A single number with no variable
   • A type of equation with no terms at all
   • Only ever a straight line

2. 2. The remainder theorem states dividing f(x) by (x − a) leaves a remainder equal to:

   • f(a)
   • Zero, always
   • x itself
   • The original polynomial

3. 3. The factor theorem states that if f(a) = 0, then:

   • (x − a) is a factor of f(x)
   • (x − a) is never a factor of f(x)
   • f(x) has no factors at all
   • a must equal zero

4. 4. The factor theorem is a special case of:

   • The remainder theorem
   • A completely unrelated theorem
   • Basic addition only
   • The Pythagorean theorem

5. 5. These theorems let you test for factors without:

   • Doing full polynomial long division every time
   • Ever substituting any values at all
   • Using any algebra whatsoever
   • Knowing the polynomial's terms

6. 6. For f(x) = x³ − 3x² + 4, f(2) equals:

   • 0
   • 4
   • 8
   • -4

7. 7. Since f(2) = 0 for f(x) = x³ − 3x² + 4, this confirms:

   • (x − 2) is a factor of f(x)
   • (x − 2) is definitely not a factor of f(x)
   • f(x) has no real factors
   • x must always equal 2

8. 8. For f(x) = x³ − x² − 4, what is f(2)?

   • 0
   • 4
   • -4
   • 8

9. 9. If f(x) = x² − 5x + 6 and f(3) = 0, then dividing f(x) by (x − 3) leaves a remainder of:

   • 0
   • 3
   • 6
   • 5

10. 10. For f(x) = x³ + 2x² − x − 2, is (x − 1) a factor?

    • Yes, since f(1) = 0
    • No, since f(1) does not equal 0
    • It is impossible to determine
    • Only if x always equals 1

11. 11. For f(x) = x³ − 6x² + 11x − 6, what is the remainder when dividing by (x − 1)?

    • 0
    • 6
    • 11
    • -6

12. 12. Why is testing f(a) = 0 a faster way to check for a factor than performing full polynomial long division?

    • Substituting a single value requires far less calculation than dividing an entire polynomial expression
    • Substitution and full long division always require exactly the same amount of calculation
    • Testing f(a) = 0 never actually confirms whether something is a factor
    • Polynomial long division is always faster than simple substitution

13. 13. Why might the factor theorem be especially useful for factorising higher-degree polynomials (like cubics or quartics)?

    • It provides a systematic way to identify factors before attempting full factorisation, rather than guessing
    • The factor theorem only applies to simple linear expressions, never higher-degree polynomials
    • Higher-degree polynomials can never be factorised using any systematic method
    • Guessing randomly is always faster than using the factor theorem

14. 14. For f(x) = x³ − 2x² − 5x + 6, which of these values of a gives f(a) = 0?

    • 1
    • 2
    • 4
    • 5

15. 15. Why does confirming one factor of a cubic polynomial (using the factor theorem) simplify finding the remaining factors?

    • Dividing out the known factor reduces the polynomial to a simpler quadratic, which is easier to factorise further
    • Finding one factor has no effect on how easily the remaining factors can be found
    • A cubic polynomial can only ever have exactly one factor in total
    • Dividing out a known factor always makes the remaining polynomial more complex, not simpler

16. 16. Why might engineers or scientists use polynomial models (tested with these theorems) to represent real-world data trends?

    • Polynomials can approximate a wide range of curved relationships, and these theorems help analyse and simplify those models
    • Polynomials can only ever represent perfectly straight-line relationships
    • Real-world data trends are never modelled using polynomial functions
    • The remainder and factor theorems have no practical application outside pure mathematics

17. 17. For f(x) = 2x³ − 3x² − 11x + 6, what is f(3)?

    • 0
    • 6
    • 3
    • -6

18. 18. For f(x) = x³ − 4x, what is f(0)?

    • 0
    • 4
    • -4
    • 1

19. 19. A polynomial of degree 3 (a cubic) can have at most how many linear factors?

    • 3
    • 1
    • 0
    • 6

20. 20. Why might knowing all the factors of a polynomial allow you to find where its graph crosses the x-axis?

    • Each factor of the form (x − a) corresponds to a point where the polynomial equals zero, which is where the graph crosses the x-axis
    • Factors of a polynomial have no relationship to where its graph crosses any axis
    • A polynomial's graph never crosses the x-axis at any point related to its factors
    • Only the leading term of a polynomial determines where the graph crosses the x-axis

21. 21. Why might the remainder theorem still be useful even when (x − a) turns out not to be a factor of a polynomial?

    • It still tells you the exact remainder from that division, which can be useful information in its own right
    • The remainder theorem only ever provides useful information when the remainder is exactly zero
    • A non-zero remainder means the theorem has failed and gives no useful information
    • The remainder theorem cannot be applied at all unless (x − a) is confirmed to be a factor