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Unit 2, Lesson 1

Unit 2, Lesson 1. Algebraic Expressions. Algebraic Expressions Defined: Monomial, Binomial, Trinomial and Polynomial.

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Unit 2, Lesson 1

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  1. Unit 2, Lesson 1 Algebraic Expressions

  2. Algebraic Expressions Defined:Monomial, Binomial, Trinomial and Polynomial An algebraic expression simply means variables and numbers (or constants) combined with operations (add, subtract, mult, divide, powers, roots). Expressions aren’t equations but two expressions can be equal to each other to make an equation A monomial is an algebraic expression (numbers and variables) that uses multiplication and the variables(s) are only in the numerator 7x, -3y2, ab3c7 are all monomials (1/x) or x-2 are not monomials b/c either the variable is in the denom or would be if there was not a negative exponent

  3. Algebraic Expressions Defined:Monomial, Binomial, Trinomial and Polynomial A binomial (bi- means two; a bi-cycle has two wheels) is two monomials connected by add. or subtraction 7x + 8; -4y2 – 6; and + 11 are all binomials A trinomial (tri- means three; tri-cycle) is three monomials connected by add or sub X2 + 6x + 9 is a trinomial A polynomial (poly- means many; a polyglot is someone who speaks many languages) means 4 or more monomials connected

  4. Adding Polynomials Adding polynomials is really the same as combining like terms (like terms have the same variables raised to the same powers) Often times polynomials are separated by parentheses: (4x – 7) + (-5x + 3)  this isn’t FOILing or distribution; the binomials are being added, not multiplied You do use distribution to get rid of the parentheses before adding though. In this case, since there is no number in front of either set of parentheses, you distribute 1: 1(4x – 7) + 1(-5x + 3) = 4x - 7 – 5x + 3 = -1x -4

  5. Adding Polynomials Another example: 2(x2- 3) + (3x2 – 5) Distribute the 2 for the first parentheses and 1 for the second parentheses: 2(x2 - 3) + 1(3x2 – 5) = 2x2 - 6 + 3x2 – 5 = 5x2 -11 Note: To help keep your signs correct, don’t subtract, always add the opposite. So for the above, 2x2 - 6 + 3x2 – 5 = 2x2 + (-6) + 3x2 + (-5)

  6. Subtracting Polynomials Subtracting polynomials is basically the same as adding polynomials with ONE difference: Instead of distributing a 1 to remove the parentheses, distribute a -1 (5y + 3) – (7y – 8) is same as 1(5y + 3) + -1(7y – 8)  5y + 3 -7y + 8 = -2y +11 Why is it a positive 8 and not still a negative 8? Another example: -2(6g + 2) - 3(g – 1)  distribute the -2 & the -3 -12g - 4 -3g + 3 = -12g + (-4) + (-3g) + 3 = -15g -1

  7. Multiplying Polynomials Notice the difference between: (2z + 4) + (-3z - 2) and (2z + 4)(-3z - 2) The first one is addition and the second one is multiplication b/c the parentheses are next to each other Multiplying polynomials is completed by multiplying all of the monomials in the first factor with each monomial in the second factor (Factors are the “things” being multiplied)

  8. Multiplying Polynomials For multiplying two binomials such as (2z + 4)(-3z - 2), FOIL (first, outer, inner, last) (2z)(-3z) + (2z)(-2) + (4)(-3z) + (4)(-2) = -6z2 + (-4z) + (-12z) + (-8) = -6z2 -16z -8 If there are more than two terms, still multiply all the monomials by each other (distribute as often as needed) (2x+ 3)(4x2 + 5x +1) = (2x)(4x2) + (2x)(5x) + (2x)(1) + (3)(4x2) + (3)(5x) + (3)(1) = 8x3 + 10x2 + 2x + 12x2 + 15x + 3 = 8x3 + 22x2 + 17x + 3

  9. Special Products:Difference of Squares Become very familiar with these. They will be used for most of the year: Difference of Squares: (a – b)(a + b) = a2 – b2 Ex: (a – 4)(a + 4) = a2 – 42 = a2 – 16 Ex: (x + 5)(x – 5) = x2 – 52 = x2 – 25 Note: It doesn’t matter if the (a – b) binomial is first or second. The key is the terms (the a & the b) are the same in both binomials and one is adding and one is subtracting

  10. Special Products:Perfect Square Trinomials For these, the key is there is a binomial that is being squared. The binomial, in other words is the base. For example: (a + b)2 = (a + b)(a + b) = a∙a + ab + ab + bb This simplifies to a2 + 2ab + b2. The short cut to do this is to take the first term (a) and square it; take the second term (b) and square it, and multiply the first and second terms (a & b) to each other and then multiply by two Ex. (x + 7)2 = x2 + 72 + (2)(7)(x) = x2 + 49 +14x = x2 + 14x +49

  11. Special Products:Perfect Square Trinomials A similar pattern occurs if the terms are subtracted instead of added. The only change is the “ab” terms are negative: (a - b)2 = (a - b)(a - b) = a∙a - ab - ab + bb Ex. (x – 6)2 = (x)(x) + (-6)(-6) + (2)(-6)(x) = x2 + 36 + (-12x) = x2 - 12x + 36

  12. Perfect Square Trinomials:Practice Without foiling, multiply the binomials: (b – 4)2 = b2 – 8b + 16 (d + 8)2 = d2 + 16d +64 (m + 1)2 = m2 + 2m + 1 (2x – y)2 4x2 – 4xy + y2Check the pattern: square the first term, square the second term, and multiple the terms together and by 2

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