Hints for solving the problems(7.3)

Recognizing patterns

These factoring problems are concerned with perfect squares and perfect cubes. What are perfect squares and perfect cubes?

Perfect squares are numbers that when you take the square root of it you get a whole number.

Perfect cubes are numbers that when you take the cube root of it you get a whole number.

Some perfect squares and perfect cubes.

       No.      Perfect Square       Perfect Cube

         1                    1                               1

         2                    4                               8

         3                    9                               27

         4                   16                              64

         5                   25                             125

         6                   36                              216

         7                   49                              343

         8                   64                              512

         9                   81                               729

        10                 100                             1000

Did you notice that when you take the square root of any number in the second column, you get the number on the left in the No. column.

Did you also notice that when you take the cube root of any number in the third column, you get the number on the left in the No. column.

Now how about a perfect square and a perfect cube that is a variable.

       Did you know that:         and   

This means that any variable with an even exponent is a perfect square and any variable with a exponent that is a multiple of three is a perfect cube.

All you do is take ½ of the exponent and eliminate the square root sign.

Remember when a perfect square, as a number, is under the radical, take the square root of it, but when a number, whether it is a perfect square or not, is under the radical as an exponent of a variable, take ½ of it.

When you have a cube root, you take 1/3 of the exponent and eliminate the square root sign.

Remember when a perfect cube as, a number, is under the radical, take the cube root of it, but when a number, whether it is a perfect cube or not, is under the radical as an exponent of a variable, take 1/3 of it.

Problem No. 2 - Factor: y² + 14y + 49 This problem is a trinomial square, that will factor into a binomial squared. How do you know it is a trinomial square. Here are some questions.

    Is the first term a perfect square?    yes   

    Is the third term a perfect square?  yes   

Can you multiply the square root of the first term times the square root of the third term and double it to get the middle term. Yes 2(y)(7) = 14y So the trinomial is a trinomial square, which factors into the square root of the first term, the sign of the middle term, and the square of the third term quantity squared. (x + 7)² the ans.

Now that wasn’t too hard.

Problem No. 5 - Factor:       4a² + 20a + 25

    square root of the first term   =   2a

    square root of the third term =    5

    multiply them together and double  2(2a)(5)  =  20a

We have a trinomial square. Remember the first and the third term has to be positive. You cannot take the square root of a negative number. The middle term can be positive or negative and that will be the sign of your binomial answer.

(2a + 5)²U the ans. Remember you can square the answer and you should get the original trinomial of the problem.

Problem No. 10 - Factor:      4c² - 28c + 49

      square root of the first term   =   2c

      square root of the third term =    7

      multiply them together and double   2(2c)(7)  =  28c

We have a trinomial square. The answer is: (2c - 7)² I changed the middle term to (-) so you would see what happened.

Problem No. 12 - Factor: m² - 144 This is the difference between two squares. How? The first term is a perfect square, (it has an exponent of 2). The second term is a perfect square,(the square root of 144 is 12). And there is a negative(-) sign between the two numbers. The factored form (the ans.) is: the square root of the first term + the square root of the second term times the square root of the first term – the square root of the second term. Wow in simple terms it is: (m + 12)(m - 12)

Problem No. 16 - Factor:      16x² - 64y²

     Is the first term a perfect square?    yes

     Is the second term a perfect square?   yes

     Is there a negative sign between the two terms?   yes

     The answer is:     (4x + 8y)(4x - 8y)

Problem No. 17 - Factor:   a³ - 216   This is the difference between two cubes. The difference or the sum of two perfect cubes can be factored. This difference of two perfect cubes will be factored into a binomial times a trinomial. You take the cube room of the first term, the sign between the two terms and the cube root of the second term: (a - 6) then you square the first term of the binomial you just derived, opposite sign of the binomial, the product of the two numbers in the binomial and add the square of the second term of the binomial: a² + (a)(6) + 6². So the answer is: (a - 6)(a² + 6a + 36) The binomial you derived must have the same sign between the two numbers as the original problem. The trinomial you derived must have a positive for the first term and the third term (when you square a number the answer is always positive). The sign of the middle term will be the opposite of the binomial in front of it. Why? Because when you multiply a binomial times a trinomial, you will get four or more terms if the middle signs are the same. By the signs being opposite, terms will become zero and there will be only two left.

Problem No. 23 - Factor:       c³ + 64

      Is the first term a perfect cube?      yes

      Is the second term a perfect cube?    yes

     The factored form is:    (c + 4)(c² - 4c + 16)

To completely factor a problem the following will be followed.

Problem No. 26 - Factor:      3a³ + 42a²b + 147ab²

To factor this problem, first factor out the GCF.

What is the GCF? -     3a

3a(a² + 14ab + 49b²) Now look at the trinomial. Is it a trinomial square? yes, so the complete factored form is:

       3a(a + 7b)²   or   3a(a + 7b)(a + 7b)

The book forgot to put an a in the third term of the original problem.

Problem No. 28 - Factor:        5x³ - 20xy²

What is the GEF? -      5x

5x(x² - 4y²) The binomial left is the difference between two perfect squares. The answer is:    5x(x + 2y)(x + 2y)