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ALGEBRA III

ALGEBRA III. Polynomials Quick Review Addition, Subtraction and Multiplication. Polynomials. A term can be a number, a variable, a product of numbers and/or variables, or a quotient of numbers and/or variables.

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ALGEBRA III

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  1. ALGEBRA III Polynomials Quick Review Addition, Subtraction and Multiplication

  2. Polynomials A term can be a number, a variable, a product of numbers and/or variables, or a quotient of numbers and/or variables. A term that is a product of constants and/or variables is called a monomial. Examples of monomials: 8, w, 24 x3y A polynomial is a monomial or a sum of monomials. Examples of polynomials: 5w + 8, 3x2 + x + 4, x, 0, 75y6

  3. A polynomial that is composed of two terms is called a binomial, whereas those composed of three terms are called trinomials. Consider the chart below with examples of different types of polynomials.

  4. Definitions Dealing with Polynomials The degree of a term of a polynomial is the number of variable factors in that term. The degree of 9x5 is 5. The part of a term that is a constant factor is the coefficient of that term. The coefficient of 4y is 4.

  5. The leading term of a polynomial is the term of highest degree. Its coefficient is called the leading coefficient and its degree is referred to as the degree of the polynomial. Consider this polynomial below: 4x2 9x3 + 6x4 + 8x  7 The termsare 4x2, 9x3, 6x4, 8x, and 7. The coefficients are 4, 9, 6, 8 and 7. The degree of each term is 2, 3, 4, 1, and 0. The leading term is 6x4 and the leading coefficientis 6. The degree of the polynomialis 4.

  6. Operations on Polynomials Polynomials can be added and subtracted by combining like terms (using the Distributive Property). Polynomials are customarily written in descending order, starting with the term with the highest power.

  7. Example Set No.1 • Combine like terms and write in descending order. • 4y4 9y4 • 7x5 + 9 + 3x2 + 6x2  13  6x5 • c) 9w5  7w3 + 11w5 + 2w3

  8. Example Set No. 1 Solutions • a) 4y4 9y4=(4 9)y4=5y4 • 7x5 + 9 + 3x2 + 6x2  13  6x5 • = 7x5 6x5+ 3x2 + 6x2 + 9  13 • = x5 + 9x2  4 • c) 9w5  7w3 + 11w5 + 2w3 • = 9w5 + 11w5  7w3 + 2w3 • = 20w5  5w3

  9. Example Set No.2 To add or subtract polynomials, perform the operation on like terms only. Simplify each expression. (a)(6x3 + 7x  2) + (5x3 + 4x2 + 3) (b)  (8x4  x3 + 9x2  2x + 72)

  10. Example Set No. 2 Solutions • (6x3 + 7x  2) + (5x3 + 4x2 + 3) • = (6 + 5)x3 + 4x2 + 7x + (2 + 3) • = x3 + 4x2 + 7x + 1 • b) (8x4  x3 + 9x2  2x + 72) • = 8x4 + x3  9x2 + 2x  72

  11. Example Set No.3 Simplify each expression. Write in descending order. a) (6x2  4x + 7)  (10x2  6x  4) b) (10x5 + 2x3  3x2 + 5)  (3x5 + 2x4  5x3  4x2)

  12. Example Set No. 3 Solutions a) 6x2  4x + 7 You can do you work (10x2  6x  4)in column form if it’s 4x2 + 2x + 11 easier. b) (10x5 + 2x3  3x2 + 5)  (3x5 + 2x4  5x3  4x2) = 10x5 + 2x3  3x2 + 5 + 3x52x4+5x3+4x2 = 13x5  2x4 + 7x3 + x2 + 5

  13. Multiplying Polynomials When multiplying a monomial by a polynomial, use the Distributive Law. 5x2(x3  4x2 + 3x  5) • = 5x5  20x4 + 15x3  25x2

  14. L F (A + B)(C + D) I O When multiplying two binomials, use the process referred to as FOIL. (A + B)(C + D) = AC + AD + BC + BD Multiply First terms: AC. Multiply Outer terms: AD. Multiply Inner terms: BC Multiply Last terms: BD ↓ FOIL

  15. Example Set No.4 Find each product. • (x + 8)(x+ 5) • (y + 4) (y 3) • (5t3 + 4t)(2t2  1) d) (4  3x)(8  5x3)

  16. Example Set No.4 Solutions a) (x + 8)(x+ 5) = x2 + 5x + 8x + 40 = x2 + 13x + 40 b) (y + 4) (y 3) = y2  3y + 4y  12 = y2 + y  12

  17. Example Set No. 4 Solution (continued) c) (5t3 + 4t)(2t2  1) = 10t5 5t3 + 8t3  4t = 10t5 + 3t3 4t d) (4  3x)(8  5x3) = 32  20x3  24x + 15x4 = 32  24x  20x3 + 15x4

  18. Example No.5 Multiply (5x3 + x2 + 4x)(x2 + 3x). Solution 1 5x3 + x2 + 4x x2 + 3x 15x4 + 3x3 + 12x2 5x5 + x4 + 4x3 5x5 + 16x4 + 7x3 + 12x2 Make sure that you have the terms lined up!

  19. Example No.5 (continued) Multiply (5x3 + x2 + 4x)(x2 + 3x). Solution 2 Distribute 5x3 Distribute 4x Distribute x2 Combine like terms.

  20. Example No.6 Multiply: (3x2 4)(2x2 3x + 1) Solution 1

  21. Example No.6 Solution Line up the like terms – leaving a space for the “missing” x term. 2x2 3x + 1 3x2 4 8x2+ 12x 4 6x4 + 9x3  3x2 6x4 + 9x3 11x2 + 12x 4

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