Identities, Worked Examples, and Step-by-Step Solutions
Special polynomial forms—such as the difference of two squares, perfect square trinomials, and the sum or difference of two cubes—provide shortcuts for factoring complex algebraic expressions. Identifying these patterns is essential for efficient problem-solving.
Factor the polynomial: \[ 16x^2 - 9y^2 \]
Rewrite terms as squares: \( 16x^2 = (4x)^2 \) and \( 9y^2 = (3y)^2 \).
\[ 16x^2 - 9y^2 = (4x)^2 - (3y)^2 \]Apply \( a^2 - b^2 = (a - b)(a + b) \):
\[ = (4x - 3y)(4x + 3y) \]Factor: \[ 4x^2 + 20xy + 25y^2 \]
Identify the squares and the middle term: \( (2x)^2 \), \( (5y)^2 \), and \( 2(2x)(5y) = 20xy \).
\[ 4x^2 + 20xy + 25y^2 = (2x)^2 + 2(2x)(5y) + (5y)^2 \]Apply \( a^2 + 2ab + b^2 = (a + b)^2 \):
\[ = (2x + 5y)^2 \]Factor: \[ 8 - 27x^3 \]
Rewrite as cubes: \( 8 = 2^3 \) and \( 27x^3 = (3x)^3 \).
\[ 8 - 27x^3 = 2^3 - (3x)^3 \]Apply \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \):
\[ = (2 - 3x)(2^2 + (2)(3x) + (3x)^2) \] \[ = (2 - 3x)(4 + 6x + 9x^2) \]Factor the following special polynomials completely.
Practice A: \( -25x^2 + 9 \)
Rearrange as a difference of squares: \( 9 - 25x^2 \).
\[ 3^2 - (5x)^2 = (3 - 5x)(3 + 5x) \]Practice B: \( 16y^4 - x^4 \)
Factor as a difference of squares twice:
\[ (4y^2)^2 - (x^2)^2 = (4y^2 - x^2)(4y^2 + x^2) \]Factor the remaining difference of squares:
\[ = (2y - x)(2y + x)(4y^2 + x^2) \]Practice C: \( \frac{1}{2}x^2 + x + \frac{1}{2} \)
Factor out \( 1/2 \):
\[ \frac{1}{2}(x^2 + 2x + 1) \]Recognize the perfect square trinomial:
\[ = \frac{1}{2}(x + 1)^2 \]Practice D: \( x^6 - 1 \)
Treat as a difference of squares: \( (x^3)^2 - 1^2 \).
\[ = (x^3 - 1)(x^3 + 1) \]Apply the difference and sum of cubes formulas:
\[ = (x - 1)(x^2 + x + 1)(x + 1)(x^2 - x + 1) \]