Use of Factor Theorem

  1. Division Algorithm for Polynomials:
    If a polynomial f(x) is divided by a non-zero polynomial g (x) then there exist unique polynomials q (x) and r (x) such that f(x)=g(x)q(x) + r(x) where either r (x) = 0 or deg r (x) < deg g(x).Here dividend = f(x), divisor = g (x), quotient = q (x) and remainder = r (x). Thus if g (x) is a quadratic polynomial, then remainder r (x) is of the form ax+b, where a, b may be zero. If divisor g (x) is a linear polynomial then r (x) is a constant polynomial, i.e., r (x) = c, where c may be zero.
  2. A non-zero polynomial g (x) is called a factor of any polynomial f (x) iff there exists some polynomial q (x) such that f(x) = g(x)q(x), i.e. iff on dividing f(x) by g (x), the remainder is zero.
  3. Remainder Theorem:
    If a polynomial f(x) is divided by (x -a), then remainder = f(a).
    If a polynomial f(x) is divided by (x +a), then remainder = f(-a).
    If a polynomial f(x) is divided by (ax +b), then remainder = f(-b/a).
  4. Factor Theorem:
    If f(x) is a polynomial and a is a real number, then (x -a) is a factor of f(x) iff f(a)=0.

Exercise

  1. Divide 2 x³ -7 x² +5 x +9 by 2 x -3 by long division method. Mention the dividend, divisor, quotient and remainder.
  2. Using remainder theorem, find the remainder when 2 x³ -7 x² +5 x +9 is divided by 2 x -3.
  3. Find the remainder (without division) on dividing f(x) by x +3, where
    (i) f(x) = 2 x² -7 x -1
    (ii) f(x) = 3 x³ -7 x² +11 x +1.
  4. Let f(x) = 2 x² -7 x -1. Find the remainder when f(x) is divided by
    (i) x -3
    (ii) 2x -3
    (iii) x /2 -3
    (iv) 2(x -3)
    (v) k (x -3), k 0
    (vi) x
    (vii) 4 x
  5. Let f(x) = 3 x³ -7 x² +1. Find the remainder when f(x) is divided by
    (i) x +2
    (ii) 2 x +2
    (iii) 1(x +2)/2
    (iv) 3(x +2)
    (v) k (x +2), k 0
    (vi) x
    (vii) 10 x
  6. Using remainder theorem, find the value of a if the division of x³ +5 x² -ax +6 by (x -1) leaves the remainder 2 a.
  7. Show that x -1 is a factor of 2 x² +x -3. Hence factorise 2 x² +x -3 completely.
  8. Show that 2 x +3 is a factor of 6 x² +5 x -6. Hence find the other factor.
  9. Show that x +2 is a factor of f(x) = x³ +2 x² -x -2. Hence factorise f(x) completely.
  10. Show that x -1 is a factor of x5 -1 while x5 +1 is not divisible by x -1.
  11. Find the constant k if 2 x -1 is a factor of f(x) = 4 x² +kx +1. Using this value of k, factorise f(x) completely.
  12. The expression 2 x³ +a x² +bx -2 leaves remainders of 7 and 0 when divided by 2 x -3 and x +2 respectively. Calculate the values of a and b. With these values of a and b, factorise the expression completely.
  13. If x +1 and x -1 are factors of f(x) = x³ +2 ax +b, calculate the values of a and b. Using these values of a and b, factorise f(x) completely.
  14. If x² -1 is a factor of f(x) = x4 +ax +b, calculate the values of a and b. Using these values of a and b, factorise f(x).
  15. Given that x² -x -2 is a factor of x³ +3 x² +ax +b, calculate the values of a and b and hence find the remaining factor.
  16. The polynomial x4 +bx³ +59 x² +cx +60 is exactly divisible by x² +4 x +3. Find the values of b and c.

Answers

1. Dividend = 2 x³ -7 x² +5 x +9, divisor = 2 x -3
   quotient = x² -2 x -1/2, remainder = -21/2
2. -21/2      3. (i) 38 (ii) -176
4. (i) -4       (ii) - 7     (iii) 29       (iv) -4     (v) -4     (vi) -1      (vii) -1
5. (i) -51     (ii) -9     (iii) -51    (iv) -51    (v) -51   (vi) 1        (vii) 1
6. 4             7. (x -1)(2 x +3)                    8. 3 x -2
9. (x -1)(x +1)(x +2)                               11. k = -4; (2 x -1)²
12. a = 3, b = -3; (x +2)(2 x +1)(x -1)
13. a = -1/2, b = 0, x (x -1)(x +1)
14. a = 0, b = -1, (x -1)(x +1)(x² +1)
15. a = -6, b = -8, third factor is x +4
16. b = 13, c = 107