MathBuster Solved Examples

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These solved examples are part of the MathBuster course of Ganit Gurooz. It covers the first part of class 9 polynomials for CBSE curriculum.

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MathBuster Solved Examples

  1. 1. MathBuster Solved Examples Lesson 02 Title: Polynomials Assignment Code: MB0902.4.1The solved examples in this assignment are based on the following concepts: 1. Definition of a polynomial in one variable, its coefficients, with examples and counter examples, its terms, zero polynomial. 2. Degree of a polynomial. Constant, linear, quadratic, cubic polynomials; 3. Monomials, binomials, trinomials.Example MB0902.4.1.01Which of the following algebraic expressions are polynomials in one variable? Give reasonsfor your answers. a. – b. c. d.MathBuster Solution StrategyFor each of the given expression we need to check two properties: 1. Is it a polynomial? 2. Is it a polynomial in one variable?Polynomials are special types of algebraic expressions which satisfy 2 conditions: 1. The denominator of a polynomial is always 1. 2. The power of each variable must be a non-negative integer.Thus, for each of the given expressions, we need to check the above two conditions. Inaddition, we also will need to check if the expressions contain only one variable.
  2. 2. Answer MB0902.4.1.01a:The given expression is – does not have a variable quantity in the denominator.We can write, – – .Thus, the first condition of it being a polynomial is satisfied.Now, there are three terms in this expression, . Thus, the power of thevariable in the three terms is 4, 1 and 0. Since none of these is negative, the expressionsatisfies the second condition of being a polynomial.Therefore, the expression – is a polynomial.Now, we observe how many variables are there in this expression. Since there is only ONEvariable, x, we conclude that – is a polynomial is one variable.Answer MB0902.4.1.01b:The given expression, has two terms, .In the second term, the variable x is in the denominator. Using the rules of exponents, we canwrite, .We can see that this term contains a negative power – of the variable. Thus, the secondcondition of being a polynomial is not met by this expression.Therefore, is not a polynomial.Answer MB0902.4.1.01c:The given expression, has no variable in the denominator of this expressionand we can write this as
  3. 3. Thus, the first condition of it being a polynomial is satisfied. Now, let’s look at each term ofthe expression. The given expression, has three terms, .Now, we can write .We can see that this term contains a fractional power of the variable. Thus, the secondcondition of being a polynomial is not met by this expression. Therefore, isnot a polynomial.Answer MB0902.4.1.01d:The given expression is does not have a variable quantity in the denominator. We canwrite, .Thus, the first condition of it being a polynomial is satisfied.Now, there are three terms in this expression, . Thus, the power of the variablein the each of the two terms is 1. Therefore, the expression satisfies the second condition ofbeing a polynomial. Therefore, the expression is a polynomial.Now, we observe how many variables are there in this expression. Since there are TWOvariables (y and z), we conclude that is NOT a polynomial in one variable.Example MB0902.4.1.02Write the coefficients of x3 in the following polynomials: a. – b. – c. d. – e. –
  4. 4. MathBuster Solution StrategyThe coefficient of a term in a polynomial is the constant factor of the term. The for the termaxn, the coefficient is a.Thus, for each polynomial given above, we must:(a) Identify the term that contains x3.(b) Identify the constant factor of this term.Answer MB0902.4.1.02:The following table provides us with the coefficients for each polynomial. Polynomial Term containing x3 Coefficient of x3 1 There is NO term containing . There is NO term containing .Example MB0902.4.1.03Write each polynomial in standard form and the determine the degree of the polynomial: a. – b. – c. d. – e. –
  5. 5. MathBuster Solution StrategyTo write a polynomial is standard form, we must combine all like terms and then arrange theterms in decreasing powers of variables. The degree of a polynomial on one variable is thehighest power of the variable in each of its terms, after the polynomial has been written instandard form.Thus, to determine the degree, we must: a. Combine all like terms of the polynomial. b. Arrange them in decreasing order of powers. c. Observe the highest power of the variable.Answer MB0902.4.1.03a:The given polynomial is – . Observe that all like terms are already combined.Rearranging the terms so that they are in decreasing power of the variable, we getThe powers of the variables in different terms are 3, 2 and 0. The highest power of thevariable is 3. Therefore, the degree of the polynomial is 3.Answer MB0902.4.1.03b:The given polynomial is – .Rearranging the terms to bring all the like terms together, we get – .Combining all the like terms, we get – .The terms are already arranged in decreasing powers of the variable. The powers of thevariables in different terms are 4, 3, 2 and 0. The highest power of the variable is 4.Therefore, the degree of the polynomial is 4.Answer MB0902.4.1.03c:The given polynomial isRearranging the terms to bring all the like terms together, we getCombining all the like terms, we get
  6. 6. The terms are already arranged in decreasing powers of the variable. The powers of thevariables in different terms are 3 and 0. The highest power of the variable is 3. Therefore, thedegree of the polynomial is 3.Answer MB0902.4.1.03d:The given polynomial is – . In this case, all the like terms are already combined and theyare arranged in decreasing order of the powers of the variable. The terms are already arrangedin decreasing powers of the variable. The powers of the variables in different terms are 2, 1and 0. The highest power of the variable is 2. Therefore, the degree of the polynomial is 2.Answer MB0902.4.1.03e:The given polynomial is – . Rearranging the terms to bring all thelike terms together, we getCombining all the like terms, we getThe terms are already arranged in decreasing powers of the variable. The powers of thevariables in different terms are 2 and 1. The highest power of the variable is 2. Therefore, thedegree of the polynomial is 2.Example MB0902.4.1.04Classify the following polynomials on the basis of the number of terms they have: a. – b. – c. d. – e. –MathBuster Solution StrategyPolynomials have special names based on the number of unlike terms they have.Monomials are polynomials with ONE term.Binomials are polynomials with TWO terms.Trinomials are polynomials with THREE terms.
  7. 7. Answer MB0902.4.1.04:The following table provides classification of the polynomials given.Polynomial Like Terms Combined Number of Terms Name 3 Trinomial 3 Trinomial 2 Binomial 1 Monomial 2 BinomialExample MB0902.4.1.05Classify the following polynomials on the basis of their degree: a. – b. – c. d. – e. –MathBuster Solution StrategyPolynomials have special names based on their degree.  Polynomials of degree ZERO are called Constant Polynomials.  Polynomials of degree ONE are called Linear Polynomials.  Polynomials of degree TWO are called Quadratic Polynomials.  Polynomials of degree THREE are called Cubic Polynomials.Answer MB0902.4.1.05:The following table provides classification of the polynomials given.Polynomial Like Terms Combined Degree Name 3 Cubic 2 Quadratic
  8. 8. 3 Cubic 2 Quadratic 1 Linear 3 0 ConstantNext Step Recommended:To see how well you have understood these problems, take the quiz MB0902.TQ.1Not Enrolled in MathBuster: Send a email to MathBuster@elipsis.in

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