3. Using factor theorem, factorize each of the following polynomials: (i) x3 – 6x2 + 3x + 10 (ii) 2y3 – 5y2 – 19y

Factorization of Polynomials Using Factor Theorem

Introduction

Factorization of polynomials is a fundamental concept in algebra that helps in simplifying expressions and solving equations. The Factor Theorem is a powerful tool that aids in factorizing polynomials.

Factor Theorem

The Factor Theorem states that if f(c)=0f(c)=0, then (x−c)(x−c) is a factor of the polynomial f(x)f(x).

Factorization of Polynomial x3−6x2+3x+10x3−6x2+3x+10

1. Step 1: Find Potential Roots

   • Potential roots can be found by setting f(x)=0f(x)=0 and solving for xx.
   • Possible rational roots are determined using the Rational Root Theorem.

2. Step 2: Test Roots Using Factor Theorem

   • Test the potential roots by substituting them into the polynomial.
   • If f(c)=0f(c)=0, then (x−c)(x−c) is a factor.

3. Step 3: Synthetic Division

   • Perform synthetic division to divide the polynomial by the found factor.
   • Repeat the process until a quadratic polynomial is obtained.

4. Step 4: Factorization

   • Factor the quadratic polynomial using methods like quadratic formula or decomposition.

Factorization of x3−6x2+3x+10x3−6x2+3x+10

1. Potential Roots:

   • Potential rational roots are ±1,±2,±5,±10±1,±2,±5,±10.

2. Testing Roots:

   • By testing, it's found that x=−2x=−2 is a root.

3. Synthetic Division:

   • Perform synthetic division:
     (x3−6x2+3x+10)÷(x+2)(x3−6x2+3x+10)÷(x+2)

   • This yields the quotient x2−8x+5x2−8x+5.

4. Factorization of Quotient:

   • The quadratic polynomial x2−8x+5x2−8x+5 can be factored as (x−5)(x−1)(x−5)(x−1).

5. Final Factorization:

   • x3−6x2+3x+10=(x+2)(x−5)(x−1)x3−6x2+3x+10=(x+2)(x−5)(x−1).

Factorization of Polynomial 2y3−5y2−19y2y3−5y2−19y

1. Potential Roots:

   • For a polynomial of the form 2y3−5y2−19y2y3−5y2−19y, potential rational roots are ±1,±12,±19,±192±1,±21,±19,±219.

2. Testing Roots:

   • By testing, it's found that y=0y=0 is a root.

3. Synthetic Division:

   • Perform synthetic division:
     (2y3−5y2−19y)÷y(2y3−5y2−19y)÷y

   • This yields the quotient 2y2−5y−192y2−5y−19.

4. Factorization of Quotient:

   • The quadratic polynomial 2y2−5y−192y2−5y−19 cannot be factored further using integer coefficients.

5. Final Factorization:

   • 2y3−5y2−19y=y(2y2−5y−19)2y3−5y2−19y=y(2y2−5y−19).

Conclusion

Factorizing polynomials using the Factor Theorem involves identifying potential roots, testing them, performing synthetic division, and factoring the resulting quotient. This method simplifies complex expressions and aids in solving polynomial equations effectively.