2 nd Semester Review

Solve Equations by Factoring Factoring Techniques GCF10y⁵ + 15y³ Difference of Squaresx² – 36 Trinomialx² – 3x – 10

Quadratic Formula How to solve Quadratic Equation. 1) Graph Difficult to do without a graphing calculator, real numbers only. 2) Factoring Only works if the trinomial/binomial is factorable, real numbers. 3) Square root Only works for x² equations without a middle term, real or imaginary numbers. 4) Quadratic Formula Can be used for any equation in the form ax² + bx + c = 0. Real or imaginary numbers.

Solving quadratic equations 1) x² - 16 = 02) x² - 6x + 2 = 0 3) x² - 16x + 64 = 04) x² - 13 = 0 5) 4x² – x = 06) 2x² – 7x + 1 =0

Properties of Square Roots How to simplify square roots. – Rewrite the radicand into prime factors. – Pairs of prime factors will be moved out of the radical. – Pairs of variables will be moved out of the radical also. – All non-pairs are left under the square root.

Square Roots of Negative Numbers Imaginary Unit (i)- negative numbers under the square root. i = √ (-1) i² = i³ = i⁴ =

Graph Quadratic Function Quadratic Function Is an equation written in the form f(x) = ax² + bx + c where a ≠ 0 Graph - is a parabola. Y-intercept- where the graph crosses the y-axis Axis of Symmetry- imaginary line where the parabola folds onto its self. Vertex-the point where the axis of symmetry and the parabola intersect. Domain: Range:

How to Graph f(x) = x² + 8x + 9 Y-intercept (0,c) Axis of Symmetry Table of Values Include the x-coordinate of the vertex Two values on each side of the vertex

Properties of Powers Negative Product of Powers Quotient of Power s Power of Power Zero Power

Polynomial What is a polynomial? No dividing of variables. No negative exponents. No variables under a radical symbol. Degree of a polynomial 1. Find the degree of each monomial(term)  Add the exponents on all the variables in the monomial 2. The largest degree is also the degree of the polynomial

Simplify Polynomials Add/subtract polynomials Combine like terms Coefficients change Variables stay the same. Be careful of subtraction signs. 1) (2a³ + 5a – 7) – (a³ – 3a + 2)

Simplifying Polynomials Multiply Polynomials 1. Distributive property or FOIL Use the rules for multiplying monomials Multiply Coefficients Add exponents 2. Combine like term 1) 2p²q(5pq – 3p³q² + 4pq⁶) 2) (y – 10)(y + 7) 3) (m – 3)²

Graphs of Polynomial Functions ConstantLinearQuadratic CubicQuarticQuintic

Examples For each graph a.Describe the end behavior b.Determine whether it represents an odd or even degree function. c.State the number of real zeros

GCF and Difference of Squares ALWAYS look for a GCF before trying other factoring techniques. It will usually make any future factoring easier. Difference of Squares a² - b² = ( + )( - ) Must be subtraction Even exponent on variables Numbers are perfect squares

Perfect Cubes Factoring perfect cubes Variable has an exponent of 3 All numbers are a perfect cubes Sum of Two cubesa³ + b³ = (a + b)(a² – ab + b²) Difference of Two cubes a³ – b³ = (a – b)(a² + ab + b²)

Trinomial Factoring ax² + bx + c = ( )( ) Two parenthesesax² + bx - c = ( )( ) Factors of the first term Factors of the last term Combination of factors (O & I) use equal the middle term and determine the signs.

Factoring by Grouping Factoring by grouping is a factoring method that can be used with four term polynomials. Regroup terms 1 & 2 together and 3 & 4 together. Find a GCF for each group If parentheses match you can use factoring by grouping. Write the multiplication problem as ( matching parentheses )( outside parentheses ) 1) x³ + 5x² + 2x ) x² + 3xy + 2xy² + 6y³

Factoring Techniques Whenever you factor a polynomial, always look for a GCF first! Then determine whether the resulting factor can be factored again using a different method. Determine the appropriate factoring method based on the number of terms and exponents. 2 terms3 term4 terms If none of the methods the factoring techniques will apply then the polynomial is called prime.

Division Polynomial ÷ Monomial Rewrite the division problem as individual monomial division problems and simplify. 1) 9x²y³ – 15xy² + 12xy³2) 16a⁵b³ – 20ab⁵ 3xy² 4ab⁷

Synthetic Division Polynomial ÷ Polynomial When dividing polynomials they must be written in descending order and every degree must be accounted for. 1. (x² + 7x – 30) ÷ (x – 3) 2. (3a⁴ – 6a³ – 2a² + a – 6) ÷ (a + 1)

Composition of Functions [f g](x) = f[g(x)] Note g[f(x)] is usually a different composition from above. Start with the inside function first and work your way out. Evaluate f[g(x)] and g[f(x)] 1. f(x) = 2x g(x) = 3x – 4 Given g(x) = -3x and f(x) = x² + 2x, find 4) f[g(2)] 5) g[f(4)]

Inverse Relations Two relations are inverse relations if and only if whenever one relation contains the element (a,b), the other relation contains the element (b,a). Q = {(1,2), (3,4), (5,6)}

How to Find an inverse of a function 1) Replace f(x) with y in the original equation. 2) Interchange x and y. 3) Solve for y. f(x) = - ½ x + 1

Square Root Functions Contains a square root of a variable. Domain: All real numbers ≥ x-value of the end point. Range: Graph goes up: All real numbers ≥ y-value of the endpoint. Graph goes down: All real numbers ≤ y-value of the endpoint.

How to Sketch a graph 1. Determine where the radicand = 0. This is the x-value of end point of the graph. 2) Make a table of values that starts with the value found in step 1 and numbers that are greater than it. 3) Plot the points found in the table.