word problem

Question

Please answer the following 3 questions.

1) Which method can we use to factor 2ab + 16a + 5b + 40? Justify your decision.

2) Is the factorization for 64a³ + 1 = (8a +1)(8a2 + 1)? Explain why or why not.

3) If a quadratic equation can be solved by using the quadratic formula, does this mean that the quadratic equation cannot be solved by factoring? Explain your answer

Answer 1

1)

 

Use factor by grouping.  This is because not every single term in the polynomial have a common factor.  However, a group of terms do have common factors.  Also, the coefficients 2 and 5 are prime.

 

2ab and 16a have a common factor of 2a.

5b and 40 have a common factor of 5.

 

So by rearranging the terms if necessary, we get

 

2ab + 16a + 5b + 40 =

 

2a(b + 8) + 5(b + 8)

 

 

Notice that (b+8) is the common factor of the sum of products.  The factored form is then

 

(2a + 5)(b + 8)

 

 

2)

 

No.  Because if we expand the right side using FOIL, it will not match the left side.

 

(8a + 1)(8a² + 1) = 64a³ + 8a² + 8a + 1 ≠ 64a³ + 1

 

 

3)

 

The quadratic equation can sometimes be solved by factoring.  We only use the quadratic formula when we cannot easily find numbers that add to the coefficient of b but multiply to the coefficient of c.  If you use the quadratic formula, you can convert the x values into factors.

 

If a solution is x=c,    then a factor is  (x - c)

 

The only times a quadratic equation cannot be solved by factoring is when we cannot find any integer factors.  And the solutions when using the formula are irrational or complex numbers.