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Transcript
Intermediate Algebra Test #03 Review Sheet Friday 10/22/04 Page 1 of 4 Section 5.1: Polynomial Functions as Mathematical Models 1. Definition: A monomial is an algebraic expression that is either a constant or a product of constants and one or more variables with whole number exponents. 2. Definition: A polynomial is a finite sum of monomials. 3. Definition: The degree of a monomial is the sum of the exponents of its variables. The degree of a nonzero constant is zero. 4. Definition: The degree of a polynomial is the highest degree of any monomial in it. 5. Definition: A polynomial in one variable is a function of the form P(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0. n is the degree of the polynomial. Section 5.2: Polynomials: Sums, Differences, and Products 1. To add or subtract polynomials, combine like terms. 2. Distributive property: a(b + c) = ab + ac. 3. To multiply polynomials, use distribution, or multiply each term in the first polynomial by each term in the second polynomial. a. Example: (x2 + 3x + 9)(x2 – 2x +3) = x4 –2x3 +3x2 +3x3 -6x2 +9x +9x2 -18x +27 4 3 =x +x +6x2 -9x +27 Section 5.3: General Forms and Special Products 1. General Form #1: (x + a)(x + b) = x2 + (a + b)x + ab 2. General Form #2: (ax + b)(cx + d) = acx2 + (ad + bc)x + bd 3. FOIL Method: First, Outer, Inner, Last a. Example: (2x + 3)(5x-9) = 10x2 – 18x + 15x – 27 = 10x2 – 3x – 27 4. Difference of Two Squares: (a + b)(a – b) = a2 – b2 5. Square of Sum: (a + b)2 = a2 + 2ab + b2 6. Square of Difference: (a – b)2 = a2 – 2ab + b2 Section 5.4: Factoring Out the Greatest Common Factor 1. Common Monomial Factoring: Factor out the greatest monomial (i.e., constants and variables of the highest power possible) common to all terms in a polynomial. a. Example: 3x2 – 15x4 + 81x6 = 3x2(1 – 5x2 + 27x4) b. Example: 5a(x – 2y) – 3b(x – 2y) = (5a – 3b)(x – 2y) 2. Factoring by Grouping: Grouping and factoring parts of a polynomial to factor the polynomial itself. a. Example: 3xb – 2b – 15x + 10 = 3xb – 2b – 15x + 10 = b(3x – 2) – 5(3x – 2) = (b – 5)(3x – 2) Section 5.5: Factoring Trinomials 1. Factoring x2 + qx + p: a. Use the fact that (x + a)(x + b) = x2 + (a + b)x + ab b. look for factors of p that sum to q c. Example: x2 – 4x – 12 = i. factors of 12 are: 1, 2, 3, 4, 6, 12 ii. the factors that add up to –4 are –6 and +2 iii. x2 – 4x – 12 = (x – 6)(x + 2) 2. Factoring Ax2 + Bx + C: Intermediate Algebra Test #03 Review Sheet Friday 10/22/04 Page 2 of 4 a. Use the fact that (ax + b)(cx + d) = acx2 + (ad + bc)x + bd b. list factors of A and B, then try out all the possibilities until you get it right c. Example: 2x2 – 9x – 18 = i. factors of 2 are: 1, 2 ii. factors of 18 are: 1, 2, 3, 6, 9, 18 iii. (2x – 2)(x + 9) = 2x2 + 18x – 2x – 18 = 2x2 – 16x – 18 iv. (2x + 3)(x – 6) = 2x2 – 12x + 3x – 18 = 2x2 – 9x – 18 3. Factoring Trinomials by Grouping: a. You’re on your own… 4. Factoring Trinomials by the AC Method: a. You’re on your own… 5. Factoring Using Special Products: a. Difference of Two Squares: a2 – b2 = (a + b)(a – b) i. Example: 9x2 – 81 = (3x + 3)(3x – 3) b. Square of Sum: a2 + 2ab + b2 = (a + b)2 i. Example: x2 + 12x + 36 = (x + 6)2 c. Square of Difference: a2 – 2ab + b2 = (a – b)2 i. Example: x2 - 2x + 1 = (x – 1)2 d. Difference of Cubes: a3 – b3 = (a – b)(a2 + ab + b2) i. Example: x3 – 8 = (x – 2)(x2 + 2x + 4) e. Sum of Cubes: a3 + b3 = (a + b)(a2 – ab + b2) i. Example: 27x3 + 64 = (3x + 4)(9x2 – 12x + 16) Section 5.6: Solving Polynomial Equations by Factoring 1. Zero-Product Rule: If a b = 0, then either a = 0 or b = 0. 2. To solve a (polynomial) nonlinear equation, get all terms on one side, then factor and apply the zero product rule. a. Example: x3 – 4x2 = 12x x3 – 4x2 – 12x = 0 x(x3 – 4x2 – 12)= 0 x(x – 6)(x + 2) = 0 x = 0 or x–6=0 or x+2=0 x=0 x=6 x = -2 Intermediate Algebra Test #03 Review Sheet Friday 10/22/04 Page 3 of 4 Section 5.7: Polynomial Division 1. Example: 2 x3 x 18 ? x3 2 x 2 6 x 17 x 3 2 x 3 0 x 2 x 18 2 x3 6 x 2 6x2 x 6 x 2 18 x 17 x 18 17 x 51 33 2 x3 x 18 33 2 x 2 6 x 17 x3 x3 Section 6.1: Rational Functions 1. A rational function is a function of the form f ( x) p ( x) , where p(x) and q(x) are polynomials and q ( x) q(x) 0. Section 6.2: Equivalent Fractions ak a (b, k 0) bk b x 2 y 2 ( x y )( x y ) ( x y ) a. Example: ( x y)2 ( x y)( x y) ( x y) 1. The Fundamental Principle of Fractions: Section 6.3: Multiplication and Division of Rational Expressions a c ac 1. Multiplication of Fractions: (b, d 0) b d bd a c a d 2. Division of Fractions: (b, c, d 0) b d b c x 3 ( x 3) 4 2 x 1 4( x 3)(2 x 1) a. Example: (8 x 4) 4 2 2 x 5x 3 (2 x 1)( x 3) 1 1 ( x 3)(2 x 1) Intermediate Algebra Test #03 Review Sheet Friday 10/22/04 Section 6.4: Sums and Differences of Rational Expressions 1. Addition and Subtraction of Rational Expressions: Page 4 of 4 a b ab (Note the need for a common c c c denominator) 3x 1 2 x 1 3x 1 x 1 2 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 2. Example: (3x 2 2 x 1) (2 x 2 3x 1) x 1 x 1 x2 5x 2 x 1 x 1 Section 6.5: Mixed Operations and Complex Fractions Keys: 1. Remember order of operations 2. Multiply numerator and denominator by the L.C.D. of all fractions… Section 6.6: Fractional Equations and Inequalities Key: Multiply both sides of the equation (or inequality) by the L.C.D. of all fractions… Section 6.7: Literal Equations Key: Solve these types of equations the same as you would any other type of linear or nonlinear equation. Section 6.8: Applications See previous chapters’ notes on solving word problems.