Questions and Answers #2 Polynomial Arithmetic

Questions and Answers Sheet 2 Polynomial Arithmetic Question #1 Determine whether 𝐹(𝑥) = 5𝑥 4 − 𝜋𝑥 3 + 1 2 is a polynomial. If it is, state its degree. If not, say why it is not a polynomial. If it is a polynomial, write it in standard form. Identify the leading term and the constant term. Answer: Step 1 Given function is, 1 𝐹(𝑥) = 5𝑥 4 − 𝜋𝑥 3 + 2 The objective is to determine whether F(x) is a polynomial or not. If yes then state its degree, if not then give the reason why it is not a polynomial. Step 2 F(x) is a polynomial of degree 4 with leading coefficient 5 and constant term Standard form: 𝐹(𝑥) = 5𝑥 4 − 𝜋𝑥 3 + 0𝑥 2 + 0𝑥 + Question #2 Evaluate each arithmetic series described. ∑40 𝑚=1(3𝑚 − 6) Answer: Step 1 ∑40 𝑚=1(3𝑚 − 6) 40 = ∑40 𝑚=1 3𝑚 − ∑𝑚=1 6 40 = 3 ∑𝑚=1 𝑚 − ∑40 𝑚=1 6 Step 2 Formula ∑𝑛𝑖=1 = 𝑛(𝑛+1) 2 ∑𝑛𝑛=1 1 = 1. 𝑛 40 = 3 ∑40 𝑚=1 𝑚 − ∑𝑚=1 6 =3 40𝑥(40+1) 2 − 6.40 =2460-240 =2220 Question #3 Show that f is continuous on (−∞, ∞). f(x) = 1 − 𝑥 2 if 𝑥 ≤ 1 ln(x) if x > 1 1 2 1 2 . Answer: Question #4 DISCUSS: Degrees of Sums and Products of Polynomials Make up several pairs of polynomials, then calculate the sum and product of each pair. On the basis of your experiments and observations, answer the following questions. (a) How is the degree of the product related to the degrees of the original polynomials? (b) How is the degree of the sum related to the degrees of the original polynomials? (c) Test your conclusions by finding the sum and product of the following polynomials: 2𝑥 3 + 𝑥 − 3 and −2𝑥 3 − 𝑥 + 7 Answer: Step 1 To explain the given problem Step 2 a) The degree of the product is just double of the original polynomial. Multiplying the parentheses a product involving the leading power of both will emerge. for example (𝑥 2 − 1), (𝑥 2 + 1) product of polynomial = (𝑥 2 − 1)(𝑥 2 + 1) = 𝑥4 − 1 b) The degree of a sum is atmost the largest of the degrees. But it could be smaller either (if the leading powers cancel.) for example 𝑥 2 and (−𝑥 2 + 𝑥) sum of polynomial = 𝑥2 − 𝑥2 + 𝑥 =x c) given that the two polynomials 2𝑥 3 + 𝑥 − 3 and −2𝑥 3 − 𝑥 + 7 product of polynomials = (2𝑥 3 + 𝑥 − 3)(−2𝑥 3 − 𝑥 + 7) = −4𝑥 6 − 4𝑥 4 + 20𝑥 3 − 𝑥 2 + 10𝑥 − 21 sum of polynomial = 2𝑥 3 + 𝑥 − 3 − 2𝑥 3 − 𝑥 + 7 =4 Question #5 To calculate: Polynomial equation with real coefficients that has roots 0, 𝑖√3. Answer: Formula used: (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2 Calculation: If the polynomial has real coefficients, then it's imaginary roots occur in conjugate pairs. So, a polynomial with the given root 𝑖√3 must have another root as −𝑖√3. Since each root of the equation corresponds to a factor of the polynomial, also, the roots indicate zeros of that polynomial, thus, the polynomial equation is written as, ( x − 0)[ x − ( I 3)][ x − (−I 3] = 0 (𝑥)(𝑥 − 𝑖√3)(𝑥 + 𝑖√3) = 0 Further use arithmetic rule, (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2 Here, 𝑎 = 𝑥, 𝑏 = 𝑖√3 and 𝑖 2 = −1 Now, the polynomial equation is, 2 (𝑥) (𝑥 2 − (𝑖√3) ) = 0 (𝑥)(𝑥 2 + 3) = 0 𝑥 3 + 3𝑥 = 0 Hence, the polynomial equation of given roots 0, 𝑖√3 is 𝑥 3 + 3𝑥 = 0. Question #6 Determine whether 𝑔(𝑥) = 𝑥 3 /2 − 𝑥 2 + 2 is a polynomial. If it is, state its degree. If not, say why it is not a polynomial. If it is a polynomial, write it in standard form. Identify the leading term and the constant term. Answer: Step 1 Given 𝑔(𝑥) = 𝑥3 2 − 𝑥2 + 2 Step 2 𝑔(𝑥) = 𝑥3 2 − 𝑥2 + 2 Convert element to fraction: 𝑥 2 = = 𝑥3 2 − 𝑥 2 ⋅2 2 + 𝑥22 2 ,2 = 2⋅2 2 2⋅2 2 Since the denominators are equal, combine the fractions: 𝑥3 − 𝑥2 ⋅ 2 + 2 ⋅ 2 = 2 Multiply the numbers: 2 ⋅ 2 = 4 𝑥 3 − 2𝑥 2 + 4 𝑔(𝑥) = 2 Hence, it is a polynomial with its degree 3, 𝑎 𝑐 𝑏 𝑎±𝑏 𝑐 𝑐 ± = Step 3 Now, convert 3 degree polynomial into standard form which is 𝑎𝑥 3 + 𝑏𝑥 3 + 𝑐𝑥 + 𝑑 Standard form ⇒ 𝑔(𝑥) = Leading term ⇒ 1 𝑥3 2 = −2𝑥 2 + 0𝑥 + 2 2 Constant term ⇒ 2 Question #7 To find: The operation performed in the polynomials 3x+1 and x-2 so that the final answer is 2x+3. Answer: Approach: In algebra, an expression with only the term is termed as monomial. It can be number or a variable or a product of both. When two or more monomials are combined using operations like addition and/or subtraction, the resultant expression is termed a polynomial. To perform subtraction operation between two polynomials, we need to use associative and commutative properties to group and combine like terms. Also, subtracting y from x is equivalent in adding -y with x. Given Information: Polynomials: 3x+1 and x-2 Final answer: 2x+3 Calculation: (3x+1)-(x-2) is equivalent to (3x+1)+(-x+2) To subtract the polynomial x-2 from 3x+1 , group and combine the like terms in the polynomials using the indicated operation. (3x+1)+(-x+2)=(3x-x)+(1+2) =(2x)+(3) =2x+3 Final statement: The arithmetic operation ''subtraction'' is performed in the polynomials 3x+1 and x-2 . Question #8 Determine whether 𝑔(𝑥) = 𝑥 3/2 − 𝑥 2 + 2 is a polynomial. If it is, state its degree. If not, say why it is not a polynomial. If it is a polynomial, write it in standard form. Identify the leading term and the constant term. Answer: Step 1 To determine the degree of polynomial. Step 2 Given 𝑔(𝑥) = 𝑥 3/2 − 𝑥 2 + 2 Step 3 A polynomial is an expression of terms that involves mathematically operations like addition, subtraction, division and multiplication where the each term consists of variable and constant and the variables have non negative integer exponents. The exponents of the variable x are 3/2 and 2 in the function 𝑔(𝑥) = 𝑥 3/2 − 𝑥 2 + 2. Since, x has non integer exponent 3/2 hence g(x) is not a polynomial. Question #9 To find: The product of two polynomials (t+2) and (3𝑡 2 − 𝑡 + 4) using indicated operation. Answer: Concept: Multiplication of polynomials: 1) To find the product of two polynomials, distribute each term of one polynomial, multiplying by each term of other polynomial. 2) Another method is to write such a product vertically, similar to the method used in arithmetic for multiplying whole numbers. Calculation: Given that (𝑡 + 2)(3𝑡 2 − 𝑡 + 4) To find the product of two polynomials (t+2) and (3𝑡 2 − 𝑡 + 4) , is to distribute each term of (t+2) , multiplying by each term of (3𝑡 2 − 𝑡 + 4) = 𝑡(3𝑡 2 − 𝑡 + 4) + 2(3𝑡 2 − 𝑡 + 4) Using Distributive law, = 3𝑡 3 − 𝑡 2 + 4𝑡 + 6𝑡 2 − 2𝑡 + 8 = 3𝑡 3 − 𝑡 2 + 6𝑡 2 − 2𝑡 + 4𝑡 + 8 [Grouping like terms] = 3𝑡 3 + 5𝑡 2 + 2𝑡 + 8 [adding coefficients of like terms] Final statement: Therefore, (𝑡 + 2)(3𝑡 2 − 𝑡 + 4) = 3𝑡 3 + 5𝑡 2 + 2𝑡 + 8 Question #10 To calculate: The terms of the polynomial 8𝑥 3 − 6𝑥 2 + 𝑥 − 7 Answer: Given information: The polynomial is 8𝑥 3 − 6𝑥 2 + 𝑥 − 7 . Formula used: The terms are separated by the arithmetic operation, addition signs. Calculation: Consider the polynomial. 8𝑥 3 − 6𝑥 2 + 𝑥 − 7 Since the terms are separated by addition signs, the terms are listed as follows: 8𝑥 3 , −6𝑥 2 , 𝑥, and -7 Therefore, the terms of the polynomial 8𝑥 3 − 6𝑥 2 + 𝑥 − 7 are 8𝑥 3 , −6𝑥 2 , 𝑥, and -7.