Hints for solving problems(6.2)

Adding and Subtracting

Problem No. 1 - Circle the algebraic expressions below that are polynomials.

(I will put yes or no to the right of the expression)

            2xy + 5xz              yes

                                no, because there is an x in the denominator and its value could be zero (0). Never divide by zero (0).

             9y² + 13yz - 8z²    yes

                                 No, because the x has an odd exponent and x could be a negative number. When you raise a negative number to an odd exponent you will get a negative answer. Multiplying this times 24 will get a negative answer. You can never take the square root of a negative number. Try it, put a negative number on your calculator and push the square root button.

                                   No, because when you reduce this fraction the answer is: . This puts a variable in the denominator. If that variable was zero (0), there would be a zero in the denominator and you can’t divide by zero.

Problem No. 4 – Identify each polynomial below as a monomial, a binomial, or a trinomial. Remember a monomial (1) has one term, a binomial (2) has two terms and a trinomial (3) has three terms. The plus and minus sign separates terms. Such as: x + y - z has three terms. But xyz is only one term.

       a.    25 - 6xyz - 4x     Trinomial, because it has three terms.

       b.           2xyz³           Monomial, because it has only one term. There aren’t any plus or minus signs to separate the factors into terms.

       c.      x + y - 1            Trinomial, because it has three terms.

       d.       36 - 3xyz          Binomial, because it has two terms.

       e.          32x²y            Monomial, because it has only one term.

Problem No. 5 - Find the degree of the polynomial

First let’s find the degree of a monomial. The degree of a monomial is the sum of the exponents of the variables. Such as: , the degree would be 9. For xyz³, the degree would be 5. Remember when you have an variable without an exponent, it is understood to be one (1). 2³x³y³z³ would have a degree of 9. You thought it would be 12, but 2 is not a variable so its exponent doesn’t count towards the degree. So the degree of a polynomial comes from the monomial (term) with the largest degree.

     degree             8         5        6     so the degree of the polynomial is 8. The terms with the largest degree is the degree of the whole polynomial.

     This is why:     2x has a degree of 1 (monomial)

                            x² + 2x - 1 has a degree of 2 (trinomial)

                            x³ - 3x² + 5x - 1 has a degree of 3 (expression with 4 terms)

Problem No. 9 - Evaluate     x³ + 3x² - x + 1   when   x  =  -2.

Substitute -2 in the expression where ever you see an x.

                           (-2)³ + 3(-2)² - (-2) + 1

                            -8    +  3(4) +   2   + 1

                             -8   +   12  +    2 + 1    =     -8 + 15    =    7  ans.

Problem No. 14 - Find:   (3x² + 7x) + (x² - 5) Drop the parenthesis and combine like terms. Remember if there is a negative sign in from of the parenthesis, all terms in the parenthesis to the right of the negative sign will change signs. Such as: -(x + 2y) will become -x - 2y.

                            3x² + 7x + x² - 5         dropping each parenthesis                      

                            4x² + 7x - 5                 combining like terms &this is the answer.

Problem No. 22 - Find:    

Remove the parenthesis and combine like terms.

You can use the commutative property to move the terms around, so like terms are next to each other. Such as:

                                    +         0            +   7abc     +    3

                                                 The answer

Problem No. 28 - Find:      

Remove the parenthesis. Remember to change the signs of all three terms in the parenthesis to the right of the negative sign.

                                           The answer

Multiplying and Dividing

Problem No. 29 - Find: Multiply the numerical coefficients and multiply the variables (remember to add exponents).

                             the answer

Problem No. 34 - Find:      

                               The answer

Problem No. 37 - Find: The monomial in the front is multiplied by all the terms in the parenthesis. Remember when the factor out front is negative every term in the parenthesis must change signs.

                                    +             +  

                                             The answer

Problem No. 46 - Find: When dividing, reduce the numerical coefficients and subtract the exponents of the like variables.

                      The answer

Problem No. 50 - Find:     

                       The answer

Problem No. 51 - Find: This is a polynomial divided by a monomial. The divisor (denominator) must be divided into each term in the dividend (numerator).

                     The answer

It is very possible to have a fraction in the answer. If the denominator has a greater degree than the numerator, there will be a fraction. This means the exponent in the denominator was larger than the numerator.

Problem No. 56 - Find:        

                            The answer