Video Notes - Algebra: Factoring, Cubics, Fractions, and Linear Equations

Practice questions covering factoring (difference of squares, perfect square trinomials, and cubes), zero product property, and solving linear equations with fractions and decimals.

What is the factorization of a^2 - b^2?

(a + b)(a - b) (Difference of Squares)

Factor 9x^2 - 16.

(3x + 4)(3x - 4)

What pattern represents a perfect square trinomial?

a^2 ± 2ab + b^2 = (a ± b)^2 (e.g., a^2 + 2ab + b^2 or a^2 - 2ab + b^2)

Factor 25a^2 - 16.

(5a - 4)(5a + 4)

Factor 27x^3 - 8.

(3x - 2)(9x^2 + 6x + 4) (Difference of Cubes)

Factor 125x^3 + 1.

(5x + 1)(25x^2 - 5x + 1) (Sum of Cubes)

What is the zero product property?

If ab = 0, then a = 0 or b = 0

Factor a^3 - b^3.

(a - b)(a^2 + ab + b^2)

Factor a^3 + b^3.

(a + b)(a^2 - ab + b^2)

Solve 5x + 5 = 10x - 20.

x = 5

Solve 4x - 8 + 6x = 72.

x = 8

Solve 1.6(x - 6) = -9.8 + 3.6x.

x = 0.1

What is the LCD method for fractions?

Multiply both sides by the least common denominator to clear fractions, then solve.

Solve -3(1 - 4) = -17 - 2n.

n = -13

Factor 49x^2 - 1.

(7x + 1)(7x - 1)

Give an example of a perfect square trinomial.

x^2 - 2x + 1 = (x - 1)^2 (pattern a^2 - 2ab + b^2)

What is the result of expanding (3x - 2)(9x^2 + 6x + 4)?

27x^3 - 8

What should you do after moving all terms to one side when solving a quadratic?

Factor and use the zero product property to find solutions.

What is a common first step when factoring a polynomial with a GCF?

Factor out the greatest common factor (GCF) from all terms.

If (ax + b)(cx + d) = 0, what are the solutions?

ax + b = 0 or cx + d = 0

What is the sum/difference of cubes factorization pattern?

a^3 ± b^3 = (a ± b)(a^2 ∓ ab + b^2)

How can you recognize a perfect square leading to a squared binomial like 3(x + 2)^2?

When the quadratic factors to a square, e.g., 3(x^2 + 4x + 4) = 3(x + 2)^2.