11. Factoring Polynomials

Notes:

Factors of polynomials

• If we multiply two polynomials together, we obtain another polynomial as the result. For example, 2x²(x – 4) = 2x³ – 8x².
• A factor is a polynomial that divides evenly into the polynomial result. So, 2x² and x – 4 are factors of 2x³ – 8x².
• Going in reverse to find the factors that multiply together to create the polynomial result is called factoring the polynomial.

Factoring a polynomial

We typically start by trying to find a common factor of each term in the polynomial, if there is one. In this case, 2x² is a common factor of 2x³ and -8x²:

• 2x³ = 2x²(x)
• -8x² = 2x²(-4)

Therefore, 2x³ – 8x² = 2x²(x – 4).

Factoring is useful for finding when the polynomial equals zero, in this case when x = 0 or x = 4. Another example: 3x³ – 3x² + 6x = 3x(x² – x + 2).

Factoring a quadratic polynomial

• Degree 2 polynomials are called quadratics. For example, let’s try to factor x² + 5x + 6 as (x + a)(x + b). We need to find numbers a and b such that a + b = 5 and ab = 6. Two values that work are 3 and 2. Check: (x + 3)(x + 2) = x² + 5x + 6.
• Another example: x² + 3x – 10 = (x + 5)(x – 2).

Factoring a difference of squares

• When multiplying (4x + 3) by its conjugate (4x – 3) the result is 16x² – 9 = (4x)² – 3². This gives a quick way to factor a difference of squares.
• For example, 9x² – 4 = (3x)² – 2² = (3x + 2)(3x – 2).
• Another example: 4x² – 25 = (2x + 5)(2x – 5).

Factoring using radicals

• We’ve considered only examples with integer coefficients up to now. But sometimes it is also possible to factor using radicals, for example, x² – 5 = (x + √5)(x – √5).

Factoring polynomials with more than one variable

• Also, we’ve considered only examples with one variable up to now. But it is also possible to factor polynomials with more than one variable, for example, 6x³y – 15xy² = 3xy(2x² – 5y).

More examples

• 2x(x – 4) + 3(x – 4) = (2x + 3)(x – 4).
• 2x³(x – 3) + 2x(x – 3)² = 2x(x – 3)(x² + x – 3).
• 2x³ – 2x² – 24x = 2x(x² – x – 12) = 2x(x + 3)(x – 4).
• 3x³y – 12xy³ = 3xy(x² – 4y²) = 3xy(x + 2y)(x – 2y).
• 3x² + 9x – 12 = 3(x² + 3x – 4) = 3(x + 4)(x – 1).
• x³ + 2x² – 9x – 18 = (x³ – 9x) + (2x² – 18)
  = x(x² – 9) + 2(x² – 9) = (x + 2)(x² – 9) = (x + 2)(x + 3)(x – 3).