Questions and Answers #3 Polynomial Arithmetic

Questions and Answers Sheet 3 Polynomial Arithmetic Question #1 To find: The product of two polynomials (8y-7) and (2𝑦 2 + 7𝑦 − 3) 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 (8𝑦 − 7)(2𝑦 2 + 7𝑦 − 3) To find the product of two polynomials (8y-7) and (2𝑦 2 + 7𝑦 − 3) , is to distribute each term of (8y-7) , multiplying by each term of (2𝑦 2 + 7𝑦 − 3) = 8𝑦(2𝑦 2 + 7𝑦 − 3) − 7(2𝑦 2 + 7𝑦 − 3) = 16𝑦 3 + 56𝑦 2 − 24𝑦 − 14𝑦 2 − 49𝑦 + 21 [Distributive law] = 16𝑦 3 + (56𝑦 2 − 14𝑦 2 ) + (−24𝑦 − 49𝑦) + 21 [Grouping like terms] = 16𝑦 3 + 42𝑦 2 − 73𝑦 + 21 [adding coefficients of like terms] Final statement: Therefore, (8𝑦 − 7)(2𝑦 2 + 7𝑦 − 3) = 16𝑦 3 + 42𝑦 2 − 73𝑦 + 21 Question #2 Assume the following sequence is arithmetic: 7, 12, ⋯ a) Create a table for this sequence x y b) What is the common difference? What is zero term? c) Write the equation for y=? Answer: Step 1 y=ax+b at x=1, y=7 1) 7=a+b x=2, y=12 2) 12=2a+b (2)-(1) a=5 7 =5+𝑏 ⇒ 𝑏 =2 y=5x+2 Step 2 x 1 2 3 4 5 6 a) y 7 12 17 22 27 32 b) Commin difference =12-7=5 zero term 𝑥=0⇒𝑦=2 c) y=5x+2 Question #3 Assume the following sequence is arithmetic: 3,6, ⋯ a) Create a table for this sequence x y b) What is the common difference? What is zero term? c) Write the equation for y=? Answer: Step 1 Just observe that this sequence consists multiples of 3. Step 2 x 1 2 3 4 5 6 a) y 3 6 9 12 15 18 b) common difference =3 zero term =0 c) y=3x Question #4 To calculate: The terms of the polynomial −𝑎2 𝑏 + 4𝑎2 − 8𝑏 + 17. Answer: Given information: The polynomial −𝑎2 𝑏 + 4𝑎2 − 8𝑏 + 17 Formula used: The terms are separed by the arithmetic operation, addition signs. Calculation: Consider the polynomial. −𝑎2 𝑏 + 4𝑎2 − 8𝑏 + 17 Since the terms are separated by addition signs, the terms are listed as follows: −𝑎2 𝑏, 4𝑎2 , −8𝑏, and 17 Therefore the terms of the polynomial −𝑎2 𝑏 + 4𝑎2 − 8𝑏 + 17 are −𝑎2 𝑏, 4𝑎2 , −8𝑏, and 17. Question #5 To calculate: Polynomial equation with real coefficients that has roots -2,i . Answer: Given: -2,i 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 must have another root as -i. 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 − (−2)](𝑥 − 𝑖)[𝑥 − (−𝑖)] = 0 (𝑥 + 2)(𝑥 − 𝑖)(𝑥 + 𝑖) = 0 Further use arithmetic rule, (𝑎 + 𝑏)(𝑎 − 𝑏) = 𝑎2 − 𝑏 2 Here, a=x, b=i and 𝑖 2 = −1. Now, the polynomial equation is, (𝑥 + 2)(𝑥 2 − (𝑖)2 ) = 0 (𝑥 + 2)(𝑥 2 + 1) = 0 𝑥 3 + 𝑥 + 2𝑥 2 + 2 = 0 𝑥 3 + 2𝑥 2 + 𝑥 + 2 = 0 Hence, the polynomial equation of given roots -2,i is 𝑥 3 + 2𝑥 2 + 𝑥 + 2 = 0. Question #6 Identify each expression as a polynomial or not a polynomial. For each polynomial, give the degree and identify it as a monomial, binomial, trinomial, or none of these. −5𝑥 11 Answer: Step 1 The given expression is −5𝑥 11 A polynomial is the combination of algebraic terms. The terms can be constant, variables or the product of the constant and the product. If the number of terms in the polynomial is one, then the polynomial is called monomial. Hence, the given expression is a monomial. Step 2 The highest exponent of the variable of the expression is called the degree of the polynomial. The highest exponent of the variable of the given expression is 11. Hence, the degree of the given monomial is 11. Question #7 Determine whether the function is a polynomial function. R(x) = 8x + 6 polynomial not a polynomial If it is a polynomial function, identify the leading coefficient, the constant term, and the degree. (If it is not a polynomial function, enter DNE for all three answers.) (a) leading coefficient (b) constant term (c) the degree Answer: Step 1 Consider the function, R(x) = 8x+6 A polynomial is an expression that can be written in the form 𝑎𝑛 𝑥 𝑛 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 1 + 𝑎0 where each real number 𝑎𝑖 is called coefficient and 𝑎0 is called a constant. Step 2 Therefore, the given function R(x) = 8x+6 is polynomial. The highest power of x is 1, so the degree is 1. The leading term is the term with the highest power i.e., 8x, and its coefficient is called the leading coefficient i.e., 8. The number that is not multiplied by a variable is called a constant. i.e., 6 Therefore, a. leading coefficient: 8 b. constant term:6 c. the degree:1 Question #8 Find specific terms of an arithmetic sequence given other terms. How many terms are there in the following finite arithmetic sequence? 7,3, −1, ⋯ , −89 1) 23 2) 24 3) 25 4) 28 5) 30 Answer: Step 1 Given arithmetic sequence, 7,3,-1,........, -89 n-th term formula is: 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑 Where, 𝑎1 = first term of the sequence = 7 𝑎𝑛 = last term of the sequence = −89 𝑑 = common difference = (𝑎2 − 𝑎1 ) = (3 − 7) = 4 𝑛 = number of terms =? Step 2 Substitute all values in the formula and solve for n, -89=7+(n-1)(-4) -89-7=-4(n-1) -96=-4(n-1) (𝑛 − 1) = (−96) (−4) (n-1)=24 n=25 Therefore, there are 25 terms in the sequence. Question #9 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, 𝐹(𝑥) = 5𝑥 4 − 𝜋𝑥 3 + 1 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 Yes, 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 + 1 1 2 . 2 Question #10 Consider the polynomial 2𝑥 3 − 3𝑥 2 + 𝑥 5 . Which of the following statements about the polynomial are not correct? A. The standard form of the polynomial is 𝑥 5 + 2𝑥 3 − 3𝑥 2 . B. The polynomial is a trinomial. C. The polynomial is quintic. D. The polynomial has a linear term. Answer: Step 1 A. Standard form of polynomialStandard form of polynomial is written with highest degree term as first term, second highest degree term as second term and so on. For the given polynomial, the standard form is 𝑥 5 + 2𝑥 3 − 3𝑥 2 Step 2 B. Check whether correct or notTrinomial is a type of polynomial which has 3 terms. The given polynomial is a trinomial as it has 3 terms, i.e., 𝑥 5 − 1 − 𝑠𝑡 term 2𝑥 3 − 2 − 𝑛𝑑 term 3𝑥 2 − 3 − 𝑟𝑑 term Step 3 C. Quintic polynomialThe polynomials having degree 5th are known as quintic polynomials. Since, the polynomial 𝑥 5 + 2𝑥 3 − 3𝑥 2 have degree =5 Hence, it is quintic polynomial. Step 4 D. Linear termLinear term is term which have degree =1 . Since, the polynomial 𝑥 5 + 2𝑥 3 − 3𝑥 2 has no term having 5th degree, hence the statement is not correct.