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To find all roots of $f(x)=0$ in a calculus framework, first analyze the function’s algebraic structure. If $f$ is a polynomial, attempt factorization: extract common factors, apply identities, or use rational root testing to reduce the degree. For transcendental or mixed forms, isolate simple components when possible (e.g., convert products to sums via logarithms only on intervals where $f>0$), or use substitutions to simplify the equation. Establish existence and count of real roots using continuity and monotonicity. If $f$ is continuous on $[a,b]$ and $f(a)f(b)<0$, the Intermediate Value Theorem __________ _____ __________ ___ ______ _______ _______ _____.
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