Chapter 8.8, Problem 1P

a.

To determine

To calculate: The factored form the given expression

Answer to Problem 1P

  $\left(4t+7\right)\left(2t^2+5\right)$

Explanation of Solution

Given: Expression is given as $8t^3+14t^2+20t+35$

Calculation:

Since the quadratic expression is in the complex form therefore we use grouping method which is most applicable method to be used here and is solved as following-

  $8t^3+14t^2+20t+35$

As we see the expression then we can solve it by splitting it into two parts and factor out greatest common factor which is shown as

  $\left(8t^3+14t^2\right)+\left(20t+35\right)$

  $2t^2\left(4t+7\right)+5\left(4t+7\right)$

Now take the factor of $\left(4t+7\right)$ and we get the required result

  $\left(4t+7\right)\left(2t^2+5\right)$

b.

To determine

To show: explaining the reason about how the factors used in the above expression.

Explanation of Solution

Given information: the expression is given as $8t^3+14t^2+20t+35$

Verification:

An expression is in factored form only if its expression specify its product.Factoring is a useful tool in solving higher degree equations as it can  change an expression from a sum or difference of terms to a product of factors.

The given polynomial is of the form $8t^3+14t^2+20t+35$ which can be factored by the best applicable method that is grouping method and by greatest common factor.

In the given expression after clear observation we first apply the grouping method in which the given polynomial is grouped into pairs as shown above.

Now find the greatest common factor from the grouped data to factorise it. This process is also known as inverse of distributive law.

Hence after solving it we get the desired result.