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J. Baker February 2010 1 • Monomials: a number, variable, or the product of quotient of a number and variable. • Polynomial: a monomial or the sum of 2 or more monomials – Types of Polynomials • Monomials (1 term) • Binomials (2 terms) • Trinomials (3 terms) BEWARE: a quotient with a variable in the denominator is NOT a polynomial! Example: A B or 2 x2 J. Baker February 2010 2 Examples Monomials NOT monomials 12 X -4x2 11ab 1/ xyz2 3 A+b 5 – 7d + 2f 5 w2 Classify the following in a table on the next page: 4x – 7 -9 X2 +2xy +y2 14m 2abc 2 + 13x 5 3x2 -11xy A + 2b + 4c 2x + 9y 3y2 3a – 7b2 -4c J. Baker February 2010 2x2 + 5x + 4 3 Classify by number of terms Monomial Binomial Degree of a monomial: Hint: variable By themselves Have a degree Of 1 Trinomial + The sum of the exponents of its variables 8y3 3 4y2 a1b1 4 -14 0 42a1b1c1 J. Baker February 2010 3 4 Degree of a polynomial with more than one term: The GREATEST degree of all terms Polynomial degree of terms degree of polynomial 3x2 + 8a2b + 4 2, 3, 0 3 7x4 + 4x - 9x2y7 4, 1, 9 9 J. Baker February 2010 5 Ordering Polynomials: the terms are written in ascending or descending order with respect to one variable’s exponents Ascending: exponents of the specified variable go from smallest to largest (up) Example: 1. 2. Descending: exponents of specified variable go from largest to smallest (down) Example: 1. 2. J. Baker February 2010 6 Adding polynomials: add the coefficients of like terms and keep the variables and exponents the same Parenthesis are not necessary in this step, the simply separate terms to be added 1, (4x + 6y) + (3x + 9y) = 7x + 15y 2. (3x2 – 5xy + 8y2) + (2x2 + xy – 6y2) = 5x2 – 4xy + 2y2 3. (3p2 – 2p + 3) = (p2 + 7) = 4p2 – 2p + 10 J. Baker February 2010 7 Subtracting Polynomials: * Distribute the negative to everything behind it * change the “ – “ to “ + “ and add as before (7x2 – 8) – (-3x2 +1) = 7x2 – 8 + 3x2 -1 = 10x2 – 9 (2a2 – ab + b2) – (3a2 + 5ab – 7ab2) = 2a2 – ab + b2 – 3a2 – 5ab + 7ab2 (x2 + y2) – (-x2 + y2) = -a2 -6ab + b2 + 7ab2 = x2 + y 2 + x2 – y2 = 2x2 J. Baker February 2010 8 Finding Perimeter: add all sides (add coefficients and leave variables the same) Square -3x2 + -3x2 + -3x2 + -3x2 = -12x2 -3x2 -m + 5n 2a + -6b2 + -3a + b2 -6b2 2m – 4n = -a -5b2 2a 2m - 4n + -m + 5n + 2m - 4n + -m + 5n -3a + b2 = 2m + 2n J. Baker February 2010 9 Review: monomial = 1 term (no + or - ) 5 2x coefficient exponent variable Laws of Exponents Product of Power: “multiplying with exponents” General rule: am * an = am+n 1. 2. 3. 4. 5. 6. 7. b3 * b8 = x (x3) (x4) = (a2b) (a5b4) = (6c2d3) (5c4d) = (10w2) (-7w2y3) = (a2b) (3a5b6) = (-ab2) (5a2) (-b3) = b11 x8 a7b5 30c6d4 -70w4y3 3a7b7 3-b5 J. Baker February-5a 2010 Add the exponents 10 Power of a Power General rule: (am)n = amn [multiply the exponents] 1. 2. 3. 4. (x5)2 = (a2)6 = (b)8 = (c5)4 = x10 a12 b8 c20 Power of a Product General Rule: (ab)m = ambm (backwards distributing) 1. (ab)4 = 2. (c2d3)5 = 3. (-2x3)4 = 4. (-9ax3y2)3 = a4b4 c10d15 12 16x J. Baker February 2010 -729a3x9y6 11 Mix it Up! Use all your rules! 1. 2. 3. 4. 5. 6. 7. (2d)2(5d3) -2x(6x2)3 (-3ab)3(2b3) (3x)2(4x2yz6) (3a)(-a2b)2(4ab2c) -5(2p2)3 (-3ab)(-3ab3)(-3a2b4)3 J. Baker February 2010 12 Quotient of Powers: dividing with exponents General rule: am = am-n an X5 5-2 = X3 = X X2 4x9 x3 50x5y3 -25x2y = 50 x5-2 y3-1 -25 = -2x3y2 = 4x9-3 = 4x6 m4w3 4-1 w3-2 = m3w = m mw2 10m4 30m Subtract the exponents = 10 m4-1 = 1 m3 30 3 J. Baker February 2010 Reminder: variables that stand alone have an exponent of 1 13 Zero Exponent B4 = B4-4 = B0 = 1 B4 Think of it as 5 or -10 5 -10 or B4 B4 General Rule: a0 = 1 ANYTHING to the zero power = 1 Negative Exponent N3 N7 = n-4 ***cannot have negative exponents*** To remove a negative exponent move only the number or variable up or down General Rule: a-n = 1 or aJ.nBaker February 2010 1 = bn b-n 14 Mix it up! 1. 3x-2 5. (6a-1b)2 (b2)4 2. -5a2b-3 3. 5a-1 10b-2 6. 24w3t4 6w7t2 7. 15x3 5x0 4. 15a-3 45a-2 J. Baker February 2010 15 Multiplying Monomial x Polynomial Distribute monomial into polynomial – Multiply coefficients – Add exponents 3(2x - 5) = 3*2x – 3*5 = 6x -15 7b( 4b2 – 18) 3d(4d2-8d-15) 7b( 4b2 – 18) = (7*4 b1+2) – (7b * 18) = 28b3 – 126b 3d(4d2-8d-15) = (3*4 d1+2) – (3*8 d1+1) – (3*15 d1) = 12d3 – 24d2 – 45d J. Baker February 2010 16 Binomial * Binomial FOIL F first: multiply the first monomials in each set of () (A + B)(C + D) A*C O outside: multiply monomials (A*C) + (A*D) + (B*C) +(B*D) on the outside of each set (). A*D I inside: multiply monomials on the inside of each set (). (2x +3)(4x +4) B*C L last: multiply monomials in the (2x * 4x) + (2x * 4) + (3 * 4x) + (3 * 4) last position in each set (). B*D Combine like terms if possible J. Baker February 2010 8x2 + 8x + 12x + 12 8x2 + 20x + 12 17 Binomial * Binomial Box Place binomials on either side of the box and multiply each box (Similar to Punnett squares in biology) 4x (2x + 3) (4x + 4) 2x +3 8x2 12x 1 2 8x 12 3 4 Combine like terms (boxes 2 and 3) +4 8x2 + 8x + 12x + 12 8x2 + 20x + 12 J. Baker February 2010 18 Polynomials: COEFFICIENTS rules to remember (number in front of variable) ADDing Add coefficients of like terms Leave exponents and variables alone. They are used only for combining like terms in this step SUBTRACTing First set of ( ) never change Distribute the negative to everything behind it Change “-” into “+” and add as normal Leave exponents and variables alone. They are used only for combining like terms in this step MULTIPLYing Multiply numbers as normal Add exponents of like terms DIVIDing Divide numbers as normal Subtract exponents of like terms. No Negative exponents allowed Exponents Raise coefficient to the power shown outside parenthesis J. Baker February 2010 -distribute the negative -add (outside parentheses) EXPONENTS Multiply exponent inside parenthesis with the exponent outside the 19 parenthesis