Use of Factor Theorem
- 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.
- 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.
- 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).
- 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
- Divide 2 x³ -7 x² +5
x +9 by 2 x -3 by long division method. Mention the dividend, divisor, quotient and remainder.
- Using remainder theorem, find the remainder when 2 x³ -7
x² +5 x +9 is divided by 2 x -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.
- 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
- 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
- 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.
- Show that x -1 is a factor of 2 x² +x -3. Hence factorise 2 x² +x -3 completely.
- Show that 2 x +3 is a factor of 6 x² +5 x -6. Hence find the other factor.
- Show that x +2 is a factor of f(x) = x³ +2 x² -x -2. Hence factorise f(x) completely.
- Show that x -1 is a factor of x5 -1 while x5 +1 is not divisible by x -1.
- 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.
- 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.
- 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.
- 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).
- 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.
- 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