Factoring

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 a2 – 7a + 10.

Solution: P (a) = a2 – 7a + 10 is divided by (a - 2).

r = p (2) = 22 – 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– 7a + 10.

Example 2: Using factorization, determine whether b - 4 is a factor of b3 – 3b2 + 4b - 12.

Solution: p (b) = b3 – 3b2 + 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 b3 – 3b2 + 4b – 12.

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Factoring Strategy

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Solving Quadratic Equations by Factoring and Applications

Quadratic equation is a polynomial equation of second degree hence it is also called as second degree equation since maximum power of variable is two. There may be zero or one or two solutions of a quadratic equation when we solve it.

General form of a quadratic equation can be expressed as:

p x2 + qx + r = 0,