Algebra factoring works on basis of factor theorem that can be called as a consequence of remainder theorem. Factor theorem gives factoring definition as follows:
If q(y) is a polynomial in variable 'y', which when divided by y- b leaves the remainder = q (b) = zero, then (y - b) is said to be a factor of q(y). Factor definition can be given as follows: A value that will satisfy expression, making it zero. Now let’s see the proof of theorem as follows:
When y(y) is divided by y - b,
R = q (b) (is the value we get by remainder theorem)
q(y) = (y - b). p(y) + q(b),
Where, p (y) is the quotient.
(Dividend = Divisor * quotient + Remainder, is what we have in our division algorithm)
But we have been given that q(b) = 0.
Hence q(y) = (y - b). p(y),
This proves that (y – b) is a factor of q(y). Conversely if y - b is a factor of q (y) then q (b) = 0.
Example 1: Determine whether (a - 2) is a factor of a^{2} – 7a + 10.
Solution: P (a) = a^{2} – 7a + 10 is divided by (a - 2).
r = p (2) = 2^{2} – 7 * 2 + 10 = 4 – 14 + 10 = 0, r = 0 i.e. calculated remainder is zero. This proves that a – 2 or a = 2 is a factor of a^{2 }– 7a + 10.
Example 2: Using factorization, determine whether b - 4 is a factor of b^{3} – 3b^{2} + 4b - 12.
Solution: p (b) = b^{3} – 3b^{2} + 4b - 12 is divided by b - 4.
R = p (4) = 64 - 48 + 16 -12 = 20, which is not equals to zero. So, b = 4 is not a factor of b^{3} – 3b^{2} + 4b – 12.
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