Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 1 Algebraic Notation The ability to convert worded sentences and problems into algebraic symbols and to understand algebraic notation is essential in the problem solving process. Notice that: • Is an algebraic expression , whereas ͬ2 + 3ͬ • Is an equation , and ͬ2 + 3ͬ = 8 • Is an inequality or inequation. ͬ2 + 3ͬ > 28 When we simplify repeated sums , we use product notation : For example: and ͬ + ͬ + ͬ = 2 ‘lots’ ͬof + ͬ = 3 ‘lots’ of ͬ ͬ = 2 × =3 × ͬ ͬ = 2 =3 ͬ ͬ When we simplify repeated products , we use index notation : For example: and ͬ × ͬ = ͬͦ ͬ × ͬ × ͬ = ͬͧ EXAMPLE 1 Write, in words, the meaning of: a) b) c) ͦ ͬ − 5 ͕ + ͖ 3ͬ + 7 a) Is “5 less than x” b) Is “the sum of a and b” or “b more than a” c) Is “7 more than three times the square of x” EXAMPLE 2 Write the following as algebraic expressions: a) The sum of p and the b) The square of the sum of p c) b less than double a square of q and q a) ͤ + ͥͦ b) (ͤ + ͥ)ͦ c) 2͕ − ͖ Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 2 EXAMPLE 3 Write, in sentence form, the meaning of: a) ̾ = ͗ͨ b) ͮ ̻ = ͦ a) D is equal to the product of c and t b) A is equal to a half of the sum of b and c, or, A is the average of b and c. EXAMPLE 4 Write “S is the sum of a and the product of g and t” as an equation. The product of g and t is ͛ͨ The sum of a and is ͛ͨ ͕ + ͛ͨ So, the equation is ͍ = ͕ + ͛ͨ TO PRACTICE EXERCISE 1 Write in words, the meaning of: a. b. c. d. ͦ 2͕ ͤͥ √͡ ͕ e. f. b+c g. h. ͕ − 3 2ͬ + ͗ (2͕)ͦ i. j. k. l. 2͕ͦ ͕ − ͗ͦ ͕ + ͖ͦ (a + b)ͦ EXERCISE 2 Write the following as algebraic expressions: a. The sum of and i. The difference between p and q if p>q · ¹ b. The sum of p, q and r j. a less than the square of b c. The product of a and b k. Half the sum of a and b d. The sum of r and the square of s l. The sum of “a” and a quarter of “b” e. The square of the sum of r and s m. A quarter of the sum of a and b f. The sum of the squares of r and s n. The square root of the sum of m and n g. The sum of twice a and b o. The square root of the sum of the squares of x and y h. The sum of x and its reciprocal Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 3 EXERCISE 3 Write in sentence form: a. ͮ c. ͆ = ͕ + ͖ b. ͇ = 3͘ ͅ = ͦ d. e. f. ͈ = ͖͗ ͎ = ͖͗ͦ ̀ = ͕͡ h. ͮͮ g. ) ͦ ͦ i. ͗ = √͕ + ͖ ̻ = ͧ ͅ = ǯ/ EXERCISE 4 Write the following as algebraic equations: a. S is the sum of p and r b. D is the difference between a and b where b>a The difference c. A is the average of k and m between two numbers is the d. M is the sum of a and its reciprocal larger one minus e. K is the sum of t and the square of s the smaller one f. N is the product of g and h g. Y is the sum of x and the square of d and e Algebraic Substitution To evaluate an algebraic expression, we substitute numerical values for the unknown, then calculate the result. Consider the number Input x crunching machine 5x - 7 alongside: calculator Output If we place any number into the machine, it calculates . So, is multiplied by 5, and Î ̎Î − ̐ Î then 7 is subtracted: For example: if and if ; ͬ = 2 5ͬ − 7 ͬ = −2 5ͬ − 7 = = 5 × 2 − 7 5 × (−2) − 7 − = = 10 7 −10 − 7 = = 3 −17 Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 4 Notice that when we substitute a negative number such as −2, we place it in brackets. This helps us to get the sign of each term correct. TO PRACTICE EXERCISE 5 If , and evaluate: ͫ = 3 ͬ = 1 ͭ = −2, a. 4 b. 4ͮ2 c. ͧ3ͯ4 d. ͩ2ͯͦ3 2 3 2 4ͯ3 e. 4ͯ3ͮ2 f. 34 ͮ2 g. 3ͯ24 h. 4 ͦ(4ͯ2) 4ͯ3 4ͮ2ͯͦ3 3 − ͫ EXERCISE 6 If , and evaluate: ͕ = −3 ͖ = −4 ͗ = −1, a. b. c. d. ͗ͦ ͖ͧ ͕ͦ + ͖ͦ (͕ + ͖)ͦ e. f. g. h. ͖ͧ + ͗ͧ (͖ + ͗)ͧ 2͕ͦ (2͕)ͦ EXERCISE 7 If , and evaluate: ͤ = 4 ͥ = −1 ͦ = 2, a. b. c. d. ǭͤ + ͥ ǭͤ + ͥ ǭͦ − ͥ ǭͤ − ͤͥ e. f. g. h. ǭͤͦ − ͥ ǭͤͦ + ͥͦ ǭͤ + ͦ + 2ͥ ǭ2ͥ − 5ͦ Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 5 DEFINITIONS: product, factors, sum, terms A product is an expression where the last operation is multiplication. In a product, the things being multiplied are called the factors . A sum is an expression where the last operation is addition. In a sum, the things being added are called the terms . As an example, consider then here is the order that computations would be done: the expression 1. Add b and c . 2. Multiply this sum by a . ͕(͖ + ͗) Notice that the last operation done is multiplication. Thus, the expression is a product. ͕(͖ + ͗) The factors are and . ͕ ͖ + ͗ As a second example, here is the order that computations would be done: consider the expression 1. Multiply a and b . 2. Add this result to c . ͕͖ + ͗ . Notice that the last operation done is addition. Thus, is a sum . ͕͖ + ͗ The terms are and . ͕͖ ͗ EXAMPLES The expression is a product . 3ͬͭ The factors are ̌, Î, Ï The expression is a product . −4ͬ(ͬ + 2) The factors are −̍, Î, Î + ̋ The expression is a sum . 5ͬ − ͭ + 1 The terms are ̎Î, −Ï, ̊ The expression 2 3 is a sum . ͬ + 2ͭ − 7 The terms are 2 3 Î , ̋Ï , −̐ EXERCISE 8 In the following expressions, how many terms are there? And each term has how many factors? a) 2͕ + 4͕͖ + 5͕(͖ + ͗) b) Ƴͬ + ͭƷͮ + 2͕ Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 6 Algebraic expressions An algebraic expression is an expression that contains one or more numbers, one or more variables, and one or more arithmetic operations. It doesn't include an equal sign. Algebraic expressions can be or many forms, for example: 3x 2 − 2x −1 1 3x 2 + 2x 3x 2 + 2x + − 6x 2 y − 2 x x 2 − 3 A term consist of products of numbers and letters, so 3x 2, -2x, -1, -6x 2y, etc. are terms The number multiplying the letters is the coefficient of the term. 3x 2 (x 2 term. Coefficient is 3) -2x (x term. Coefficient is -2) -1 (constant term is -1) Polynomials Polynomials are algebraic expresions. A polynomial in is a sum of terms, each of the form k ͬ , where:͕ͬ is a real number, ͕ is a nonnegative integer. That is, ͟ ͟ = {0 ,1 ,2,3 ,...} . DEFINI TION: standard form of polynomials ; degree; leading coefficient The standard form of a polynomial is: n n-1 n n-1 1 0 · Î + · Î +. + · Î + · Here, n denotes the highest power to which is raised; this highest power is ͬ called the degree of the polynomial. Thus, in standard form, the highest power term is listed first, and subsequent powers are listed in decreasing order. Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 7 Notice that in the notation i (read as " "), the ͕iͬ · É˸ ¿ Ê¿Ã»É Î ÊŠʾ» ¿ number denotes the coefficient of the i term. ͕i ͬ The number , which is the coefficient of the highest power term, is called ͕n the leading coefficient of the polynomial. Note that a constant (like 5 ) can be written as 5x 0 . This is why the power is allowed to equal zero in the definition of polynomial—to allow for constant terms. EXAMPLE 5 The expression 5x 4-x3-3x 2+7x-5 is a polynomial. Find its terms, coefficients, leading coefficient and degree. The terms are: 5x 4, -x3, -3x 2, 7x, and -5 . Comparing each term with the required form ax k , we have: writing in the term a k form ax k 5x 4 5x 4 a=5 k=4 -x3 (-1)x 3 a= -1 k=3 -3x 2 -3x 2 a= -3 k=2 7x 7x a=7 k=1 -5 -5x 0 a= -5 k=0 Notice that every value of a is a real number , and every value of k is a nonnegative integer. The standard form of this polynomial is: 4 3 2 5ͬ −ͬ −3ͬ +7ͬ − 5 Here, the highest power term is written first, and subsequent terms decrease in power. The degree is 4 , since this is the highest power.