Math 2: Algebra 2, Geometry and Statistics Ms. Sheppard-Brick 617.596.4133 http://lps.lexingtonma.org/Page/2434 Name: Date: Chapter 2 Quiz 1 Review Sheet Students Will Be Able To: • Use basic rules of algebra to transform expressions. (TB 2.02, 2.04) • Determine whether expressions are equivalent. (TB 2.02) • Factor expressions by finding a common factor. (TB 2.03) • Apply the zeroproduct property to factored expressions. (TB 2.03, 2.04) • Recognize and provide examples of polynomials. (TB 2.06) • Use the terms coefficient and degree to solve problems (TB 2.06) • Expand polynomials and express them in normal form (TB 2.07) • Determine whether polynomials in different forms are equivalent (TB 2.07) • Add, subtract, and multiply polynomials (TB 2.08) COMPLETE THIS ASSIGNMENT ON A SEPARATE SHEET OF PAPER!! 1. Write a polynomial that fits each description a. A cubic binomial with a quadratic term with a coefficient of 5. b. A quadratic trinomial in standard form with a constant of −3. c. A quartic monomial with a coefficient of 7. d. A quadratic binomial with no linear term. 2. Use the polynomial ! + − 6! a. Show that − 2+ 3= ! + − 6! b. Let x = 6. Find all the solutions to ! + − 6! = 0 (I.e. solve for y.) 3. Use the zero product property (ZPP). Find all the solutions to each equation. a. − 2 + 1 = 0 b. 2− 3 + 5 = 0 c. 3+ 8 = 0 d. ! − 4= 0 4. Find the greatest common factor of each set of expressions. a. 8! and 12b. 24! and 16! c. ! ! , ! , and ! ! d. 10(+ 2) ! and 15(+ 2) ! 5. Solve for x in each equation. Use factoring and ZPP. a. 3! + 9! = 0 b. 4! = 8! c. ! = 49d. ! = 49e. − 2 + − 2 = 0 f. + 2+ + 2= 0 6. Transform the expression below into standard form. Then find all the values of a that make the coefficient of the linear x term 0. (! + 3+ )(! + 3− 7)
• Use basic rules of algebra to transform expressions. (TB 2.02, 2.04) • Determine whether expressions are equivalent. (TB 2.02) • Factor expressions by finding a common factor. (TB 2.03) • Apply the zero-‐product property to factored expressions. (TB 2.03, 2.04) • Recognize and provide examples of polynomials. (TB 2.06) • Use the terms coefficient and degree to solve problems (TB 2.06) • Expand polynomials and express them in normal form (TB 2.07) • Determine whether polynomials in different forms are equivalent (TB 2.07) • Add, subtract, and multiply polynomials (TB 2.08)
COMPLETE THIS ASSIGNMENT ON A SEPARATE SHEET OF PAPER!!
1. Write a polynomial that fits each description a. A cubic binomial with a quadratic term with a coefficient of 5. b. A quadratic trinomial in standard form with a constant of −3. c. A quartic monomial with a coefficient of 7. d. A quadratic binomial with no linear term.
2. Use the polynomial 𝑥! + 𝑥𝑦 − 6𝑦!
a. Show that 𝑥 − 2𝑦 𝑥 + 3𝑦 = 𝑥! + 𝑥𝑦 − 6𝑦! b. Let x = 6. Find all the solutions to 𝑥! + 𝑥𝑦 − 6𝑦! = 0 (I.e. solve for y.)
3. Use the zero product property (ZPP). Find all the solutions to each equation.
a. 𝑥 − 2 𝑥 + 1 = 0 b. 2𝑥 − 3 𝑥 + 5 = 0 c. 3𝑥 𝑥 + 8 = 0 d. 𝑥! − 4𝑥 = 0
4. Find the greatest common factor of each set of expressions.
a. 8𝑥! and 12𝑥 b. 24𝑥! and 16𝑥! c. 𝑎!𝑏!, 𝑎!𝑏, and 𝑎!𝑏!𝑐 d. 10(𝑥 + 2)! and 15(𝑥 + 2)!
5. Solve for x in each equation. Use factoring and ZPP.
a. 3𝑥! + 9𝑥! = 0 b. 4𝑥! = 8𝑥! c. 𝑥! = 49𝑥 d. 𝑥! = 49𝑥 e. 𝑎 𝑥 − 2 + 𝑏 𝑥 − 2 = 0 f. 𝑥 𝑎 + 2𝑏 + 𝑎 + 2𝑏 = 0
6. Transform the expression below into standard form. Then find all the values of a
that make the coefficient of the linear x term 0. (𝑥! + 3𝑥 + 𝑎)(𝑥! + 3𝑥 − 7)
8. A square is (𝑥 + 1) inches on each side. You cut a smaller square hole from the
larger square that is (𝑦 + 1) inches on each side. In terms of x and y, find the area of the leftover shape in square inches. Include a diagram.
9. Write each expression as a product of two expressions. One expression should be the greatest common of the terms. If there is no common factor, state that as your answer.
a. 10𝑥! − 18𝑥! b. 18𝑝! − 27𝑞! c. 𝑝𝑞 + 𝑝𝑟 d. 𝑚!𝑛 −𝑚𝑛!
e. 5𝑥 4𝑥 − 1 − 6 4𝑥 − 1 f. 2𝑥 − 3 2𝑥 + 2𝑥 − 3 5 g. 4𝑝 𝑥 − 3 + 7𝑞 𝑥 − 3 h. 𝑝 𝑟 + 𝑠 + (𝑟 + 𝑠)
10. Find the value of d such that 𝑥 + 𝑑 𝑥 + 5 = 𝑥! + 8𝑥 + 15
11. Find two polynomials that make each condition true.
a. The polynomials have the same degree and a sum of 𝑥! + 5𝑥 + 3 b. The polynomials have a different degree and a sum of 𝑥! + 5𝑥 + 3 c. The polynomials have a sum with degree 3 and a product with degree 5. d. The polynomials have a sum with degree 4 and a product with degree 6. e. The polynomials have a sum of −10. f. The polynomials have a product of 𝑥! − 16
12. Use ℎ 𝑥 = 𝑥! − 3𝑥 + 8
a. Find a polynomial 𝑘 𝑥 such that ℎ 𝑥 + 𝑘 𝑥 = 3𝑥! + 5𝑥 + 10 b. Find a polynomial 𝑚 𝑥 such that ℎ 𝑥 +𝑚 𝑥 = 2𝑥! − 3 c. Find a polynomial 𝑛 𝑥 such that ℎ 𝑥 + 𝑛 𝑥 has degree 2 and ℎ 𝑥 ∗ 𝑛 𝑥
has degree 3.
13. Find the product of (𝑥 + 𝑥! + 𝑥!) and (𝑥! + 𝑥! + 𝑥! + 𝑥!)
14. Consider the polynomial 𝑝 𝑥 = 3𝑥! − 8𝑥 + 𝑥! − 5+ 4𝑥! − 2𝑥! + 𝑥! − 8𝑥! a. Write 𝑝(𝑥) in normal form. b. What is the degree of the polynomial? c. What is the coefficient of the cubic term? d. What is the coefficient of the quadratic term? e. Find a polynomial that you can add to 𝑝(𝑥) to get a polynomial with degree 5.
