1. **Stating the problem:** We need to understand and perform calculations involving polynomials and monomials. 2. **Definitions:** - A **monomial** is an algebraic expression consisting of one term, for example, $3x^2$ or $-5y$. - A **polynomial** is a sum of one or more monomials, for example, $2x^3 + 4x - 7$. 3. **Important rules:** - When multiplying monomials, multiply the coefficients and add the exponents of like bases: $$a x^m \times b x^n = (a \times b) x^{m+n}$$ - When adding or subtracting polynomials, combine like terms (terms with the same variable and exponent). 4. **Example calculation:** Multiply the monomials $3x^2$ and $4x^3$. 5. **Step-by-step:** - Multiply coefficients: $3 \times 4 = 12$ - Add exponents of $x$: $2 + 3 = 5$ - Result: $$12x^5$$ 6. **Example calculation:** Add the polynomials $2x^3 + 4x - 7$ and $x^3 - 2x + 5$. 7. **Step-by-step:** - Combine like terms: - $2x^3 + x^3 = 3x^3$ - $4x - 2x = 2x$ - $-7 + 5 = -2$ - Result: $$3x^3 + 2x - 2$$