Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

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1 Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial Degree Simplify each sum. Name Using Degree Number of Terms Name Using Number of Terms constant 1 monomial linear 2 binomial 2 quadratic 3 trinomial 3 cubic 1 monomial 4 fourth degree 2 binomial Name each expression based on its degree and number of terms Write like terms in columns and perform necessary operation. Simplify each sum or difference Changing subtraction to addion of the inverse 8. Page1of7 SBMC2/16/09

2 Multiplying and Factoring Algebra Standards 10.0 and 11.0 Operations with Monomials and Polynomials Example A: Simplify the expression : To solve exponent problems with coefficients, separate the variables from coefficients. 9. Simplify the expression. Separate the coefficients from variables. Multiply and divide the coefficients. Combine the x terms in the numerator and denominator. Use product of power. 10. Simplify the expression. Use Quotient of Powers Simplify. 11. Simplify the expression. Example B: Simplify the expression. Use Power of Power. 12. A triangle has a base of length and a height of. What is the area of the triangle? Simplify Use Product of Power Simplify 13. The sum of two polynomials is. One polynomial is What is the other? Page2of7 SBMC2/16/09

3 Factoring Trinomials, Perfect of Squares and Differences of Squares Algebra Standards 10.0 and 11.0 Area models are used to illustrate products of two binomials. We can use these models when factoring. Since and, there are two possible arrangements to try. Study the area models below. When the rectangle is completed in each model, there are 11 rectangles on the left and 7 rectangles with area x in the model on the right, the most on the right demonstrates the correct factorization:. Remember the FOIL method? It is useful when factoring polynomials such as. F O I L (n + 2)(n + 5) = = Difference of Two Squares ( ) Example C: Factor Trinomial Squares Example E: Factor Write a square Look for binomial of the from (a b) (a b) Find two numbers whose sum is 8 and whose product is 15. Write as a difference of two squares. Product of 15 Sum Trinomial Squares Example D: Factor = The sign of the middle term is positive. = Notice, +3 and +5 gives the sum of 8 and have a produce of 15. So, substitute +3 and +5 for b. Therefore, = (x + 3) (x + 5) Page3of7 SBMC2/16/09

4 Solving Quadratic Equations Algebra Standards 10.0, 14.0, 20.0, 21.00, 22.0 and 23.0 When solving, you must first determine if you can factor greatest common term. If you can easily recognize a difference of two squares or a perfect square or a regular square trinomial then factor and solve. If not, complete the square. If you cannot complete the square, use quadratic formula. Completing the Square when is 1. Example F: Solve Notice: there are no common factors; it is neither a perfect square trinomial, nor a difference of two squares. It looks like a regular square trinomial, but cannot be factored using those methods, so let s complete the square to solve. Given equation Move original constant to the other side. Add new constant to both sides (the square of half the coefficient of x) Write left side as perfect square Zz Square root both sides (remember to use plus-or-minus): Solve for x. Notes: Find all real roots. Factoring can only find integer or rational roots. When you write it as a binomial squared, the constant in the binomial will be half of the coefficient of x Observations and New Learning: Page4of7 SBMC2/16/09

5 Solving Quadratic Equations Example G: Solve. 14. Solve Take the square root on both sides 15. Solve Example H: Find the roots Take the square root on both sides Solve 16. Solve 17. Solve Ex 18. Solve Example I: Solve 19. Solve Add +49 to both sides. Divide both sides by 3. Take the square root on both sides 20. Complete the square 21. Solve by completing the square Rationalize the Denominator 22. Solve by completing the square 23. Solve by completing the square 24. Solve by completing the square 25. Solve by completing the square Page5of7 SBMC2/16/09

6 Quadratic Formula Algebra Standard 19.0 Standard Form Substitute 0 for y. Divide by a so that the coefficient on is 1. Subtract from both sides of the equation. Make the left side of the equation a perfect square. Factor left side of the equation. Simplify right side. Take square root of both sides of the equation. Simplify each side. Isolate x term. Subtract from both sides of equation. Simplify right side of equation. 26. Factor 27. Factor Solve by using the quadratic formula Factor Factor 30. Factor 31. Factor Page6of7 SBMC2/16/09

7 : Algebra 1 Benchmark III Study Guide Answer Key 1 Constant, monomial 2 Not a polynomial 3 Quadratic binomial 4 Cubic trinomial 5 Cubic binomial x = 0, x = x = -12, x = x = -4, x = (x 4 (2x 1) (m + 3)(m + 7) 30 (x 6y)(x + 5y) 31 (x + 11)(x + 9) 32 x = 8, x = x = 11, x = x = -15, x = x = Page7of7 SBMC2/16/09