Algebra for CUET UG: Equations, Quadratics, Modulus, Logs & Progressions

A polynomial is an algebraic expression in which variables have only non-negative integer powers. That rule instantly removes terms like $x^{-1}$ or $\sqrt{x}$ from the polynomial category.

For a multivariable term like $7x^3y^2$, the degree is $3+2=5$. The degree of the whole polynomial is the highest such value among its terms.

A linear equation keeps the variable to power 1. Most CUET one-variable questions reduce to the same move-and-divide pattern.

Two equations in two variables are solved together because both conditions must hold at once. In MCQs, elimination is usually faster than substitution.

If the lines intersect, there is one solution. If they are parallel, there is no solution. If they overlap, there are infinitely many solutions.

A quadratic has the form $ax^2+bx+c=0$ with $a\ne 0$. Some are solved fastest by factorization, while others are better handled through root relations.

The discriminant classifies the roots without solving the full equation.

If $D>0$, roots are distinct and real. If $D=0$, roots are equal and real. If $D<0$, roots are non-real.

If a polynomial $f(x)$ is divided by $x-a$, the remainder is $f(a)$.

Watch the sign carefully: for divisor $x+3$, substitute $x=-3$.

Inequalities look like equations with direction. Their most important rule is simple: dividing or multiplying by a negative reverses the sign.

Modulus means distance from zero, so it is always non-negative. Equations like $|x-a|=k$ naturally split into two cases.

Logs turn exponent questions into compact notation. If $a^x=y$, then $\log_a y=x$.

An arithmetic progression has a constant difference between consecutive terms. Confirm the difference first, then use formulas.