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: 2 +3 2 +3=8 2 +3>28 Is an algebraic expression, whereas Is an equation, and Is an inequality or inequation. When we simplify repeated sums, we use product notation: For example: + = 2 lots of = 2 = 2 and ++ = 3 lots of =3 =3 When we simplify repeated products, we use index notation: For example: = and = EXAMPLE 1 Write, in words, the meaning of: a) 5 b) + c) 3 +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 square of q a) + b) (+) c) 2 b) The square of the sum of p and q c) b less than double a
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. 2 b. c. d. e. 3 f. b+c g. 2+ h. (2) i. 2 j. k. + l. (a+b) EXERCISE 2 Write the following as algebraic expressions: a. The sum of and b. The sum of p, q and r c. The product of a and b d. The sum of r and the square of s e. The square of the sum of r and s f. The sum of the squares of r and s g. The sum of twice a and b h. The sum of x and its reciprocal i. The difference between p and q if p>q j. a less than the square of b k. Half the sum of a and b l. The sum of a and a quarter of b m. A quarter of the sum of a and b n. The square root of the sum of m and n o. The square root of the sum of the squares of x and y
Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 3 EXERCISE 3 Write in sentence form: a. =+ b. = c. =3 d. = e. = f. = g. = h. = + i. = 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 c. A is the average of k and m d. M is the sum of a and its reciprocal e. K is the sum of t and the square of s The difference between two numbers is the larger one minus 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 crunching machine alongside: Input x 5x - 7 calculator Output If we place any number into the machine, it calculates. So, is multiplied by 5, and then 7 is subtracted: For example: if =2 5 7 = 5 2 7 = 10 7 = 3 and if = 2 ; 5 7 = 5 ( 2) 7 = 10 7 = 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 =3, =1 and = 2, evaluate: a. b. c. d. e. () f. g. h. EXERCISE 6 If = 3, = 4 and = 1, evaluate: a. b. c. + d. (+) e. + f. (+) g. 2 h. (2) EXERCISE 7 If =4, = 1 and =2, evaluate: a. + b. + c. d. e. f. + g. ++2 h. 2 5
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 the expression (+) then here is the order that computations would be done: 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, consider the expression +. here is the order that computations would be done: 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 3 is a product. The factors are,, The expression 4(+2) is a product. The factors are,, + The expression 5 + 1 is a sum. The terms are,, The expression 2 + 2 3 7 is a sum. The terms are 2, 3, EXERCISE 8 In the following expressions, how many terms are there? And each term has how many factors? a) 2 + 4 + 5( + ) b) + +2
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 + 6x 2 y 2 x 3x x 2 2 + 2x 3 A term consist of products of numbers and letters, so 3x 2,-2x,-1,-6x 2 y, 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,...}. DEFINITION: standard form of polynomials; degree; leading coefficient The standard form of a polynomial is: n n + n-1 n-1 +...+ 1 + 0 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 i (read as " "), the number i denotes the coefficient of the i term. The number n, which is the coefficient of the highest power term, is called 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 -x 3-3x 2 +7x-5 is a polynomial. Find its terms, coefficients, leading coefficient and degree. The terms are: 5x 4, -x 3, -3x 2, 7x, and -5. Comparing each term with the required form ax k, we have: term writing in the form ax k a k 5x 4 5x 4 a=5 k=4 -x 3 (-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: 5 4 3 3 2 +7 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. The leading coefficient is 5, since this is the coefficient of the highest power term. Notice that the leading coefficient actually leads (comes at the beginning of) the polynomial, WHEN the polynomial is written in standard form. EXAMPLE 6 The following expressions are NOT polynomials. Why? a) 1x + x - 1 b) x - x 1/2 c) 7x 2-7x + 7x 1/ 2
Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 8 a) 1x + x - 1 is not a polynomial; no negative powers are allowed. b) x - x 1/2 is not a polynomial; the number 1/2 is not an allowable power. c) 7x 2-7x + 7x 1/ 2 is not a polynomial; the number 1/2 is not an allowable power. DEFINITION: monomial, binomial, trinomial A polynomial with exactly one term is called a monomial. A polynomial with exactly two terms is called a binomial. A polynomial with exactly three terms is called a trinomial. DEFINITION: quadratic, cubic, quartic A polynomial of degree 1 is called linear A polynomial of degree 2 is called quadratic. A polynomial of degree 3 is called a cubic. A polynomial of degree 4 is called a quartic. Polynomials have beautiful smooth graphs no breaks and no kinks. The higher the degree of a polynomial, the more it is allowed to "turn" (change direction). Indeed, it can be shown easily (using calculus) that a polynomial of degree n can have at most n-1 turning points. The graph below is the polynomial ()= 3. Notice that this polynomial has degree 3 and has 2 turning points.
Departamento de Matematicas. Real Instituto de Jovellanos. J. F. Antona Algebraic notation and Polynomials 9 EXERCISE 9 1) What is the degree of these polynomials? a) 2x 16 +80x 8 b) 7769x-97x 7-56x 9 +31x 19 c) 21x 4-12x 11 +6x 2-1710x 10 +4171x 14 d) 60x 3 +65x 5-3425x 15 e) 8+27x 11 +234x 18 2) What name is given to a polynomial with exactly one term? 3) What is the leading coefficient of these polynomials? a) 85-54x 20 b) 100x 19-57x 20 +10029x 6 +2x 4 c) 61x 4-6x 3-7726x 12 +1425x 11 +45x 19 d) 9643x 3-45x 16-97x 19 4) A polynomial is a sum of terms, each of a particular form. What is this form? 5) Is 28 an allowable term in a polynomial? 6) Is 2954 an allowable term in a polynomial? 7) Is 7x 70 an allowable term in a polynomial? 8) Is the term x 9 an allowable term in a polynomial? 9) What is a quartic function? 10) What name is given to a polynomial with exactly two terms? 11) What is a trinomial? 12) What name is given to a polynomial with exactly one term? 13) Suppose that a polynomial has degree 8. What (if anything) can be said about the number of turning points for this polynomial? 14) Suppose that a polynomial has 5 turning points. What (if anything) can be said about the degree of this polynomial? 15) Write the following polynomial in standard form: a) -3x 14 +8x 27 +6x 5 +4x 6 b) 4x 2-7x 4 +5x 6 c) -6x 26 +3x 9-7x 6 d) 5x-2x 25-2x 2 e) 2x 3 +8x 6 +8x 7 +4x-3x 2 16) Can the graph of a polynomial have a break in it? 17) Can the graph of a polynomial have a kink in it?