Nonlinear Algebra
Five topics on nonlinear algebra, the SAT's Advanced Math domain and the densest content on the test. Exponents and radicals, expanding and factoring polynomials, quadratics and the parabola, rational, radical, and absolute-value equations, and nonlinear functions. This is where the freshman's dream lives, the belief that a power or a root spreads across a sum, and where a difference of squares quietly splits into two conjugate factors.
Topics
2.2
Polynomial Expressions and Factoring
Lesson
Diag
MC
locked
Reference The exponent, radical, factoring, quadratic, rational, and absolute-value rules behind the problems in this unit
Multiply like bases (add)
$x^{a}\cdot x^{b} = x^{a+b}$
Divide like bases (subtract)
$\dfrac{x^{a}}{x^{b}} = x^{a-b}$
Power of a power (multiply)
$(x^{a})^{b} = x^{ab}$
Power over a product, never a sum
$(xy)^{a} = x^{a}y^{a}$
Negative exponent (reciprocal)
$x^{-a} = \dfrac{1}{x^{a}}$
Fractional exponent (root)
$x^{\tfrac{m}{n}} = \sqrt[n]{x^{m}}$
Square of a sum (keep the middle term)
$(x+a)^{2} = x^{2}+2ax+a^{2}$
Difference of squares (conjugates)
$x^{2}-a^{2} = (x+a)(x-a)$
Factor a quadratic (multiply to the constant, add to the middle)
$x^{2}+(p+q)x+pq = (x+p)(x+q)$
Quadratic formula (keep the two branches and the $2a$)
$x = \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
Discriminant (two if positive, one if zero, none if negative)
$b^{2}-4ac$
Vertex form (vertex at $(h,k)$, sign inside flips)
$y = a(x-h)^{2}+k$
Axis of symmetry (the vertex x-coordinate)
$x = -\dfrac{b}{2a}$
Radical equation (square, then check every candidate in the original)
$\sqrt{f(x)} = g(x) \Rightarrow f(x) = g(x)^{2}$
Rational equation (exclude any value that makes a denominator zero)
$\dfrac{p(x)}{x-a}, \quad x \neq a$
Absolute value (two cases if the right side is positive, none if negative)
$|f(x)| = k \Rightarrow f(x) = \pm k \ \ (k > 0)$
Unit 2 tools