Overview
This lecture reviews essential Algebra 1 concepts and problem-solving techniques to help students prepare for their final exam, including systems of equations, polynomials, factoring, radicals, and quadratic equations.
Systems of Equations
- Solve systems by graphing, substitution, and elimination (addition) methods.
- The solution to a system is where all equations are true (intersection point).
Polynomials & Exponents
- Use product, power, and quotient rules for exponents.
- Combine like terms; perform polynomial addition, subtraction, and multiplication.
- FOIL method helps multiply binomials.
- Degree of a polynomial is the highest power of its variable.
- The area and volume problems may involve polynomials.
Factoring
- GCF (Greatest Common Factor): The lowest power of any common variable across all terms.
- Factor trinomials by finding pairs that multiply to the last term and add to the middle coefficient.
- Use factoring by grouping for polynomials with four terms.
- Recognize and factor the difference of squares: ( a^2 - b^2 = (a+b)(a-b) ).
- Not all polynomials can be factored; some are prime.
Solving Equations
- Set factored expressions equal to zero and solve each factor.
- Quadratic formula: ( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} ) is used when factoring is not possible.
Radicals
- Simplify radicals by factoring out perfect squares.
- Combine like radicals by adding/subtracting coefficients.
- Rationalize denominators by multiplying by the necessary radical.
- Multiply/divide expressions under the same radical when possible.
Radical Equations & Geometry
- Isolate the radical before squaring both sides to solve equations.
- Use the Pythagorean theorem (( a^2+b^2=c^2 )) to find triangle sides.
- The square root property yields two solutions: ( x^2 = a \rightarrow x = \pm\sqrt{a} ).
Quadratic Functions & Graphing
- The vertex of a parabola can be found at ( x = -\frac{b}{2a} ).
- Axis of symmetry runs through the vertex.
- Y-intercept: set ( x = 0 ); X-intercepts: set ( y = 0 ).
- Parabolas do not always cross the x-axis (real vs. imaginary roots).
Key Terms & Definitions
- System of Equations — set of two or more equations with the same variables.
- Polynomial — an algebraic expression with terms of variables with non-negative integer exponents.
- FOIL — method for multiplying two binomials: First, Outer, Inner, Last.
- GCF (Greatest Common Factor) — largest factor shared by all terms in a polynomial.
- Difference of Squares — an expression of form ( a^2 - b^2 ) factoring to ( (a+b)(a-b) ).
- Radical — an expression involving a root, such as a square root.
- Rationalize — process to eliminate radicals from denominators.
- Quadratic Equation — equation of the form ( ax^2 + bx + c = 0 ).
- Vertex — turning point of a parabola on a graph.
Action Items / Next Steps
- Practice problems on factoring, radicals, and quadratic equations.
- Review and memorize key formulas (quadratic formula, exponent rules, Pythagorean theorem).
- Graph sample quadratic functions and identify intercepts and vertex.
- Complete any assigned homework or review packets.