Factorization of Quadratic Polynomials - Methods

Factorization of Quadratic Polynomials - Methods

TL;DR

Factorizing a quadratic polynomial ( ax^2 + bx + c ) means rewriting it as a product of two linear factors, usually by splitting the middle term so the two pieces multiply to ( ac ) and add to ( b ). This article covers four methods, six worked examples, and the sign errors that derail most attempts.

What Does It Mean to Factorize a Quadratic Polynomial?

Factorizing a quadratic polynomial means expressing ( ax^2 + bx + c ) as a product of two linear factors of the form ( (px+q)(rx+s) ). When the quadratic equals zero, those factors provide the roots directly.

If ( ax^2 + bx + c = (x-h)(x-k) ), then ( h ) and ( k ) are the roots of the quadratic. Not every quadratic factors neatly over the rational numbers. When it does not, the quadratic formula still finds the roots.

How Do You Factorize a Quadratic Polynomial?

There are four standard methods:

Splitting the middle term, step by step

For ( ax^2 + bx + c ), find two numbers whose product is ( a \times c ) and whose sum is ( b ).

  1. Compute ( ac ).
  2. Find two numbers multiplying to ( ac ) and adding to ( b ).
  3. Split the middle term.
  4. Group and factor each pair.
  5. Factor out the common bracket.

Examples of Factorization of Quadratic Polynomials

Example 1: Factorize ( x^2 + 8x + 15 )

  1. ( ac = 1 \times 15 = 15 )
  2. Two numbers are 3 and 5.
  3. Split: ( x^2 + 3x + 5x + 15 )
  4. Group: ( x(x+3) + 5(x+3) )
  5. Factor: ( (x+3)(x+5) )

Final answer: ( (x+3)(x+5) )

Example 2: Factorize ( x^2 - 2x - 15 )

  1. Product is ( -15 ) and sum is ( -2 ).
  2. Correct numbers are -5 and 3.
  3. Split: ( x^2 - 5x + 3x - 15 )
  4. Group: ( x(x - 5) + 3(x - 5) )
  5. Factor: ( (x - 5)(x + 3) )

Final answer: ( (x-5)(x+3) )

Example 3: Factorize ( 6x^2 - 5x - 6 )

  1. ( ac = 6 \times (-6) = -36 )
  2. Numbers are -9 and +4.
  3. Split: ( 6x^2 - 9x + 4x - 6 )
  4. Group: ( 3x(2x - 3) + 2(2x - 3) )
  5. Factor: ( (2x - 3)(3x + 2) )

Final answer: ( (2x-3)(3x+2) )

Common Mistakes in Factorization

  1. Getting the signs of the two numbers backwards. Check both conditions: product equals ( ac ) and sum equals ( b ).
  2. Forgetting the leading coefficient when ( a \neq 1). Always use ( a \times c ) as the product target.
  3. Assuming every quadratic factors over the rationals. Check the discriminant.

Practice Questions on Factorization of Quadratic Polynomials

  1. Factorize ( x^2 + 7x + 12 ).
  2. Factorize ( x^2 - x - 12 ).
  3. Factorize ( 2x^2 + 7x + 3 ).
  4. Factorize ( x^2 - 14x + 49 ).
  5. Factorize ( 25x^2 - 9 ).
  6. Factorize ( 3x^2 + 5x - 4 ).

Answers

  1. ( (x+3)(x+4) )
  2. ( (x-4)(x+3) )
  3. ( (2x+1)(x+3) )
  4. ( (x-7)^2 )
  5. ( (5x+3)(5x-3) )
  6. No rational factors; roots are ( x = \frac{-1 \pm \sqrt{13}}{3} ).

Key Takeaways