Special Products - Factoring, 11

Category: Algebra

"Published in Suisun City, California, USA"

Find the factors for

Solution: 

Consider the given equation above

If you think that the above equation cannot be factored, then you must consider first the investigation of each terms whether they can be factored or not. The variables at the first and last terms are perfect square. Since the last term is negative, then obviously we cannot take a square root of a negative number and hence, the given equation is not a perfect trinomial square. We can check the above equation using discriminant if it can be factored or not as follows

where a, b, and c are the coefficients of a trinomial. Now, let's check the given equation as follows 

Since the value of discriminant is a whole number, then the given equation can be factored. Next, we have to think the factors of the first and last terms so that when we add the product of two factors of the first and last terms, it will be the same as the middle term. The factors of the first term are 1, 15, 3, and 5. The factors of the last term are 1, 15, 3, and 5. Since the last term is negative, then one of the two factors must be negative. We need to do the trial and error in assigning the factors as follows:

Trial 1: Use 1 and 15 for the first term and 3 and -5 for the last term.

 The middle term is (1)(-5) + (3)(15) = -5 + 45 = 40.
 
Trial 2: Use 5 and 3 for the first term and 3 and -5 for the last term.

The middle term is (5)(-5) + (3)(3) = -25 + 9 = -16.

Since the middle term is -16ab which is exactly the same as the answer above, then the factors of the given equation are

 

Special Products - Factoring, 10

Category: Algebra

"Published in Newark, California, USA"

Find the factors for 

Solution: 

Consider the given equation above

If you think that the above equation cannot be factored, then you must consider first the investigation of each terms whether they can be factored or not. The variables at the first and last terms are perfect square. Since the last term is negative, then obviously we cannot take a square root of a negative number and hence, the given equation is not a perfect trinomial square. We can check the above equation using discriminant if it can be factored or not as follows

where a, b, and c are the coefficients of a trinomial. Now, let's check the given equation as follows

Since the value of discriminant is a whole number, then the given equation can be factored. Next, we have to think the factors of the last term so that we add the two factors, it will be the same as the middle term. The possible factors of the last term are 1, -1, 135, -135, 5, -5, 27, -27, 3, -3, 45, -45, 9, -9, 15, and -15. If the middle term is -134, then the factors must be 1 and -135. When you add 1 and -135, it will give us -134. Therefore, the factors of the above equation are

Special Products - Factoring, 9

Category: Algebra

"Published in Newark, California, USA"

Find the factors for

Solution:

Consider the given equation above

The first and the last terms are both perfect squares. The middle term is twice the product of the square roots of first and last terms. Let's check the middle term as follows

Since the middle term is exactly the same as the computation above, then the given trinomial is a perfect trinomial square. If the middle term is negative, then the square root of the last term must be negative. Therefore, the factors are

  

Therefore, the factors are