MA100 Week One Summary Note

Distributive Law

  • Formula: a(x+y)=ax+aya(x+y)=ax + ay
  • Example: 5×13=5(10+3)=(5×10)+(5×3)=655 \times 13 = 5(10+3) = (5 \times 10) + (5 \times 3) = 65
  • Negative example: −(x+y)=−x−y-(x+y)=-x-y
  • Difference of squares: x2−y2=(x+y)(x−y)x^2 - y^2 = (x+y)(x-y)

Adding and Multiplying Fractions

  • Adding: xb+db=x+db\frac{x}{b} + \frac{d}{b} = \frac{x+d}{b} (only for numerators)
  • Multiplying: ab×cd=acbd\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}
  • Dividing: ab÷cd=ab×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Factoring Polynomials

  • Factoring out common terms: e.g. 3x2−6x=3x(x−2)3x^2 - 6x = 3x(x-2)
  • Product of binomials: (x+r)(x+s)=x2+(r+s)x+rs(x+r)(x+s) = x^2 + (r+s)x + rs
    • Example: (x+3)(x+5)=x2+8x+15(x+3)(x+5) = x^2 + 8x + 15

Completing the Square

  • Form: 0=(x+r)2+k0 = (x+r)^2 + k
  • Example: Solve 0=x2+2x+10 = x^2 + 2x + 1 leads to (x+1)2=0(x+1)^2 = 0

Quadratic Formula

  • Formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  • Discriminant:
    • If b2−4ac>0b^2 - 4ac > 0, two distinct real roots.
    • If b2−4ac=0b^2 - 4ac = 0, one repeated real root.
    • If b2−4ac<0b^2 - 4ac < 0, no real roots (complex roots).

Exponents and Radicals

  • Exponent properties:
    • an=a×a×… (n times)a^n = a \times a \times … \text{ (n times)}
    • a=a1/2\sqrt{a} = a^{1/2}
  • Rationalizing the denominator: example, 3xx+9−3\frac{3x}{\sqrt{x+9}-3}
    • Multiply by the conjugate: (x+9+3)(\sqrt{x+9}+3)