Hints for solving the problems(6.3)

Multiplying Binomials

Problem No. 1 - Find:      (a + 2)(a + 5)

There are three ways to multiply two binomials.

                         a + 2      multiplicand

                         a + 5       multiplier

                      5a + 10         5(a + 2)

              a² + 2a                  a(a + 5)

              a² + 7a + 10       the answer

b. Use the distributive property. Split up the first binomial and multiply each term by the second binomial.

                    a(a + 5) + 2(a + 5)

                      a² + 5a + 2a + 10

                         a² + 7a + 10      the answer

c. Foil is the method you need to know. This means you multiply the first terms together, the outer two, the inner two, and the last two. The inner two and outer two are added algebraically together.

                         (a + 2)(a + 5)

       first two      outer two      inner two      last two

            a²       +         5a       +        2a       +       10

                           a² + 7a + 10        the answer

Problem No. 8 - Find:       (3m - 4n)(7m + 2n)

       first two      outer two      inner two      last two

      (3m)(7m) +    (3m)(2n)  +   (-4n)(7m)  + (-4n)(2n)

                      21m² + 6mn - 28mn - 8n²

                            21m² - 22mn - 8n²      the answer

Problem No. 18 - Find:      (3x - 2y)(3x - 2y)

       first two      outer two      inner two      last two

        (3x)(3x)  +    (3x)(-2y)   +   (-2y)(3x)  +  (-2y)(-2y)

                           9x² - 6xy - 6xy + 4y²

                               9x² - 12xy + 4y²      the answer

This problem is the product of the same two binomials, which can be stated as:              (3x - 2y)².    The pattern for this is:   (a ± b)² = a² ± 2ab + b²

This says the square of the first term, plus or minus two times the product of the two terms, plus the square of the second term.

     (3x - 2y)²

Square the first term   (3x)²,    multiply two times the product of the two terms

      2(3x)(-2y),     and square the second term. (2y)²

                     9x² - 12xy + 4y²      the answer

Problem No. 19 - Find:     (4a - 7c)(4a - 7c)   =   (4a - 7c)²

Square the first term   (4a)²,  multiply two times the product of the terms

       2(4a)(-7c),   and square the second term. (-7c)²

                      16a² - 56ac + 49c²      the answer

Problem No. 22 - Find:   (a + 7b)(a - 7b)  This problem is the product of the sum and difference between two numbers.

          first two      outer two      inner two      last two

               a²        -        7ab     +      7ab        -      49b²

                                   a²   -   49b²      the answer

There is a short cut for this. The pattern for this is:   (a + b)(a - b) = a² - b²

To multiply the sum and different of two binomials, you square the first term minus the square of the second term.

         (a + 7b)(a - 7b)

Square the first term (a²) minus the square of the second term (49b²)

              a² - 49b²      the answer

Problem No. 25 - Find:   (5x + 3)(5x - 3)   When you square the first term and minus the square of the second term you get:

                    25x² - 9      the answer

Multiplying and Dividing

Problem No. 34 - Find:      (a + 2b)(a² + 6a - 3b)

To multiply a polynomial by a polynomial, use the distributive property. Multiply each term of the first polynomial by the second polynomial.

              a(a² + 6a - 3b) + 2b(a² + 6a - 3b)

               a³ + 6a² - 3ab + 2a²b + 12ab - 6b²  Now combine like terms and put the answer in decreasing exponents according to the alphabet.

                   a³ + 6a² + 2a²b + 9ab - 6b²    the answer

Problem No. 43 - Find:    (x³ + x² - 13x + 14) (x - 2)

To divide a polynomial by a polynomial, you set it up like a long division problem.

                                  x² + 3x - 7

Such as:           x - 2 )x³ +  x² - 13x + 14

                                -(x³ - 2x²)

                                         3x² - 13x

                                       -(3x² - 6x)

                                                 -7x + 14

To do this division, you divide the first term of the dividend (the term under the division sign) by the first term of the divisor (the term outside the division sign on the left). That is x³/x. Which is x². This goes above the first term of the dividend. Then you multiply the x² by both terms in the divisor and put that answer under the dividend. You want like terms in a column. The next operation is subtraction and you change the sign of the each of the numbers you just wrote under the dividend and use the rules of addition of signed numbers. This procedure is worked through out the problem.

These are long and hard problems. During these operations you will do addition, subtraction, multiplication, and division of monomials and polynomials.

Problem No. 50 - Find:      (2x³ + 7x² - x - 2) (2x + 1)

                             x² + 3x - 2

                 x + 1 ) 2x³ + 7x² - x - 2

                                     6x² - x

                                  -(6x² + 3x)

                                            - 4x - 2

                                           -(-4x - 2)

                                                   0        the remainder

                                         x² + 3x - 2     the answer

Problem No. 51 - Find:        (4x³ + 7x² - 14x + 6) (4x - 1)

                            x² + 2x - 3

              4x - 1 ) 4x³ + 7x² - 14x + 6

                       -(4x³  -   x²)

                                    8x² - 14x

                                  -(8x² -  2x)

                                            -12x + 6

                                          -(-12x + 3)

                                                        3      the remainder

If there is a remainder it has to be put as a fraction over the divisor in your answer.

                       x² + 2x - 3 +        the answer