Algebra (modulus and polynomials)

The absolute value that saves bridges

• Engineers designing suspension cables need to know the maximum deviation from the ideal — whether above or below.
• The modulus 绝对值 (absolute value) measures distance from zero, ignoring direction. It appears everywhere: error bounds, tolerances, and inequalities.

The modulus function

• The modulus $|x|$ is the size of a number, sign removed ($|x| \geq 0$). Its graph is a "V".
• Useful rules:
  • $|a| = |b| \Leftrightarrow a^2 = b^2$,
  • $|x - a| < b \Leftrightarrow a - b < x < a + b$.

The modulus function $y = |x - 2|$ creates a V-shape: everything below the axis is reflected above it.

Factor & remainder theorems 余数定理

• A polynomial 多项式 is a sum of powers of $x$; its degree 次数 is the highest power.
• Remainder theorem: the remainder when $p(x) \div (x - a)$ is $p(a)$.
• Factor theorem 因式定理: $(x - a)$ is a factor exactly when $p(a) = 0$.

Taking the modulus folds the negative part of the line up into a V

Worked example — factor theorem

• Show that $(x - 2)$ is a factor of $p(x) = x^3 - 5x^2 + 2x + 8$.
• $p(2) = 8 - 20 + 4 + 8 = 0$. Since $p(2) = 0$, $(x - 2)$ is a factor.
• Divide: $x^3 - 5x^2 + 2x + 8 = (x - 2)(x^2 - 3x - 4) = (x - 2)(x - 4)(x + 1)$.

Opposite signs of f(a) and f(b) trap a root between a and b

Solving polynomial equations

• Use the factor theorem to find one root, then divide to reduce the degree.
• A cubic has at most 3 real roots; a quartic at most 4.
• Sketch the graph to check: the number of $x$-axis crossings matches the number of real roots.
• Differentiate a fraction with the quotient rule.

You've got it

Write from the solutions 3 and -7

• Solve the modulus equation absolute value of x + 2 equals 5.