FACTORING POLYNOMIALS - USING FACTOR THEOREM AND SYNTHETIC DIVISION, EXAMPLES

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Please study
Algebra Factoring before Factoring Polynomials
if you have not already done so.

There, we explained the terms Factors or Divisors,
Factorization of Polynomials, Prime Polynomial etc.

That knowledge is a prerequisite here.

Here, we factorize higher degree ( > 2 ) polynomials using
Factor Theorem which is based on Remainder Theorem.











Remainder Theorem

If a Polynomial in x, say f(x) is divided by (x - a), then the remainder is f(a).

Proof :
We know Dividend = Divisor x Quotient + Remainder
Let q(x) be the Quotient and r be the Remainder when f(x) is divided by (x - a).
Then we have f(x) = (x - a) x q(x) + r where in the degree of q(x) is one less than that of f(x) and r is a constant.
Since the equation is true for all real values of x, putting x = a in the equation, we get
f(a) = (a - a) x q(a) + r = 0 + r = rr = f(a).
Thus, when f(x) is divided by (x - a), the remainder is f(a). (Proved.)
Note:

If f(x) is divided by (ax + b), then the remainder is f(-ba)

The proof for this is similar to the one given above.

Example on Remainder Theorem : Factoring Polynomials

Find the remainder when 3x3 - 2x2 + x + 2 is divided by (i) x - 1 (ii) x + 2 (iii) 2x -1 (iv) 3x + 2

Solution:
Let f(x) = 3x3 - 2x2 + x + 2
(i) when f(x) is divided by (x - 1),
The remainder = f(1) = 3(1)3 - 2(1)2 + (1) + 2 = 3 - 2 +1 + 2 = 4. Ans.
(ii) when f(x) is divided by (x + 2),
The remainder = f(-2) = 3(-2)3 - 2(-2)2 + (-2) + 2 = 3(-8) - 2(4) - 2 + 2 = -32. Ans.
(iii) when f(x) is divided by (2x -1),
The remainder = f(1⁄2) = 3(1⁄2)3 - 2(1⁄2)2 + (1⁄2) + 2 = 3(1⁄8) - 2(1⁄4) + 1⁄2 + 2 = (3 - 2 x 2 + 1 x 4 + 2 x 8)⁄8 = 19⁄8 Ans.
(iv) when f(x) is divided by (3x + 2),
The remainder = f(-2⁄3) = 3(-2⁄3)3 - 2(-2⁄3)2 + (-2⁄3) + 2 = 3(-8⁄27) - 2(4⁄9) - 2⁄3 + 2 = (3 x -8 - 2 x 4 x 3 - 2 x 9 + 2 x 27)⁄27 = -12⁄27 = -4⁄9 Ans.

An important application of Remainder Theorem is the
Factor Theorem which is applied in Factoring polynomials.





Factor Theorem : Factoring Polynomials

We know that, if the remainder is zero, when a Polynomial f(x) is divided by (x - a), then (x - a) is a factor of f(x).
But, as per remainder theorem, remainder = f(a).
∴ If f(a) = 0, then (x - a) is a factor of f(x).
Also If (x - a) is a factor of f(x), then f(a) = 0.
This is Factor theorem.

(x - a) is a factor of f(x) ⇔ f(a) = 0

Factor Theorem is very useful in Factoring
Polynomials of higher (than 2) degree .

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Method of Factoring Polynomials using Factor Theorem :

The method of Factoring Polynomials
involves the following steps.


STEP 1: Let the Polynomial be f(x). Check whether (x- a) is a factor of f(x) or not. To check whether (x - a) is a factor of f(x) or not, we have to checkwhether f(a) is zero or not. Choosing values of a is done by trial and error.To facilitate, reducing the number of trials, we take the constant termof the Polynomial, f(x), find its factors, put + and - to the factors and takethose values for a and check with them, if f(a) is zero or not. This isdone till we find an a such that f(a) is zero.
STEP 2: After deciding (x - a) is a factor, we divide the Polynomial f(x) by (x - a) to get the Quotient. This division is better done by Synthetic Divisionwhich is explained in Solved Example 1 of Set 3 below.
STEP 3: we, then Factorize the Quotient for further factors. This Factorizingis done either by Factor theorem (Repeat from Step 1) or by other methodsdepending upon the degree of the Quotient.

The method of Factoring Polynomials
will be clear by the following set of Solved Examples.

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Example 1 of Factoring Polynomials

Solve the following problem on Factoring Polynomials

If a and b are unequal and x2 + ax + b and x2 + bx + a have a common factor,show that a + b + 1 = 0

Solution to Example 10 of Factoring Polynomials

Let f(x) = x2 + ax + b; and p(x) = x2 + bx + a
Let (x - k) be the common factor of f(x) and p(x).⇒ f(k) = 0 and p(k) = 0
f(k) = 0 ⇒ k2 + ak + b = 0............(i)
p(k) = 0 ⇒ k2 + bk + a = 0............(ii)
(i) - (ii) gives (k2 + ak + b) - (k2 + bk + a) = 0 - 0 = 0
ak + b - bk - a = 0 ⇒ k(a - b) -1(a - b) = 0 ⇒ (a - b)(k- 1) = 0
⇒ ( (k- 1) = 0 [ Since by data a and b are unequal ⇒ (a - b) ≠ 0 ]
k = 1.
Substituting this value of k in (i), we get
(1)2 + a(1) + b = 0 ⇒ a + b + 1 = 0 Thus, If a and b are unequal and x2 + ax + b and x2 + bx + a have a common factor,we could show that a + b + 1 = 0 using Factor Theorem in Factoring polynomials.

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Exercise on Factoring polynomials

Solve the Following Problems on Factoring polynomials :

  1. If f(x) = x2 + 5x + a and p(x) = x2 + 3x + b have a common factorthen (i) find the common factor (ii) show that (a - b)2 = 2(3a - 5b)

For Answer See at the bottom of the Page.

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Answers to Exercise on Factoring Polynomials

Answers to the problems in Exercise
on Factoring Polynomials are given below.

  1. (i) x + (a - b)⁄2