Transcript Chapter 5
TMAT 103 Chapter 5 Factoring and Algebraic Fractions TMAT 103 §5.1 Special Products §5.1 – Special Products • a(x + y + z) = ax + ay + az • (x + y)(x – y) = x2 – y2 • (x + y)2 = x2 + 2xy +y2 • (x – y)2 = x2 – 2xy +y2 • (x + y + z)2 = x2 + y2 + z2 + 2xy + 2xz + 2yz • (x + y)3 = x3 + 3x2y + 3xy2 + y3 • (x – y)3 = x3 – 3x2y + 3xy2 – y3 TMAT 103 §5.2 Factoring Algebraic Expressions §5.2 – Factoring Algebraic Expressions • Greatest Common Factor ax + ay + az = a(x + y + z) • Examples – Factor the following 3x – 12y 40z2 + 4zx – 8z3y §5.2 – Factoring Algebraic Expressions • Difference of two perfect squares x2 – y2 = (x + y)(x – y) • Examples – Factor the following 16a2 – b2 36a2b4 – 100a4z10 256x4 – y16 §5.2 – Factoring Algebraic Expressions • General trinomials with quadratic coefficient 1 x2 + bx + c • Examples – Factor the following x2 + 8x + 15 q2 – 3q – 28 x2 + 3x – 4 2m2 – 18m + 28 b4 + 21b2 – 100 x2 + 3x + 1 §5.2 – Factoring Algebraic Expressions • Sign Patterns Equation Template x2 + bx + c ( + )( + ) x2 + bx – c ( + )( – ) x2 – bx + c ( – )( – ) x2 – bx – c ( + )( – ) §5.2 – Factoring Algebraic Expressions • General trinomials with quadratic coefficient other than 1 ax2 + bx + c • Examples – Factor the following 6m2 – 13m + 5 9x2 + 42x + 49 9c4 – 12c2y2 + 4y4 TMAT 103 §5.3 Other Forms of Factoring §5.3 – Other Forms of Factoring • Examples – Factor the following a(b + m) – c(b + m) 4x + 2y + 2cx + cy x3 – 2x2 + x – 2 36q2 – (3x – y)2 y2 + 6y + 9 – 49z4 (m – n)2 – 6(m – n) + 9 §5.3 – Other Forms of Factoring • Sum of two perfect cubes x3 + y3 = (x + y)(x2 – xy + y2) • Examples – Factor the following x3 + 64 8z3m6 + 27p9 §5.3 – Other Forms of Factoring • Difference of two perfect cubes x3 – y3 = (x – y)(x2 + xy + y2) • Examples – Factor the following m3 – 125 8z3 – 64p9s3 TMAT 103 §5.4 Equivalent Fractions §5.4 – Equivalent Fractions • A fraction is in lowest terms when its numerator and denominator have no common factors except 1 • The following are equivalent fractions a b = ax bx §5.4 – Equivalent Fractions • Examples – Reduce the following fractions to lowest terms x2 – 2x – 24 2x2 + 7x – 4 a2 – ab + 3a – 3b a2 – ab x4 – 16 x4 – 2x2 – 8 x3 – y3 x2 – y2 TMAT 103 §5.5 Multiplication and Division of Algebraic Fractions §5.5 – Multiplication and Division of Algebraic Fractions • Multiplying fractions a b • c d = ac bd . • Dividing fractions a c b d = a b • d c = ad bc . §5.5 – Multiplication and Division of Algebraic Fractions • Examples – Perform the indicated operations and simplify 4t4 • 12t2 6t 9t3 a2 – a – 2 a2 + 7a + 6 • a2 + 3a – 18 a2 – 4a + 4 4 15pq2 39mn 13m5n3 5p4q3 TMAT 103 §5.6 Addition and Subtraction of Algebraic Fractions §5.6 Addition and Subtraction of Algebraic Fractions • Finding the lowest common denominator (LCD) 1. Factor each denominator into its prime factors; that is, factor each denominator completely 2. Then the LCD is the product formed by using each of the different factors the greatest number of times that it occurs in any one of the given denominators §5.6 Addition and Subtraction of Algebraic Fractions • Examples – Find the LCD for: 2 8 , 125 , and 307 4 x , y32 , and xy52 4 x 2 6 x 9 , 3 ( x 3) , and 5 x 2 9 §5.6 Addition and Subtraction of Algebraic Fractions • Adding or subtracting fractions 1. Write each fraction as an equivalent fraction over the LCD 2. Add or subtract the numerators in the order they occur, and place this result over the LCD 3. Reduce the resulting fraction to lowest terms §5.6 Addition and Subtraction of Algebraic Fractions • Perform the indicated operations 4 s 3 1s 1 6x 2 x2 y2 1 3 x 6 1 2 x4 1 x 2 3 xy 2 y 2 3 x 2 xy 2 y 2 TMAT 103 §5.7 Complex Fractions §5.7 Complex Fractions • A complex fraction that contains a fraction in the numerator, denominator, or both. There are 2 methods to simplify a complex fraction – Method 1 • – Multiply the numerator and denominator of the complex fraction by the LCD of all fractions appearing in the numerator and denominator Method 2 • Simplify the numerator and denominator separately. Then divide the numerator by the denominator and simplify again. §5.7 Complex Fractions • Use both methods to simplify each of the complex fractions 2 1 c 2 1 c 3 5x x 2 4 3 x 2 2 TMAT 103 §5.8 Equations with Fractions §5.8 Equations with Fractions • To solve an equation with fractions: 1. Multiply both sides by the LCD 2. Check • Equations MUST BE CHECKED for extraneous solutions – – Multiplying both sides by a variable may introduce extra solutions Consider x = 3, multiply both sides by x §5.8 Equations with Fractions • Solve and check 4 x 3 9 2 x 5 x 2 x4 3 2 x Solve V Q R1 1 4 Q R2 for R 2