Chapter 8.7, Problem 46PPS

a.

To determine

To state if the given statement is true or false.

Answer to Problem 46PPS

False

Explanation of Solution

Given:

Statement: All binomials that have a perfect square in each of the two terms can be factored.

The factor of sum of squares cannot be factored in the binomial. They can only be subtracted.

For example,

  $a^2-b^2=x^2-4=(x+4)(x-4)$ which is the difference of squares.

Whereas considering the sum of squares, $a^2+b^2=x^2+4$ cannot be factorised.

Here the binomial is again with perfect square terms, but they cannot be factored.

Conclusion:

The given statement is false.

b.

To determine

To state if the given statement is true or false.

Answer to Problem 46PPS

False

Explanation of Solution

Given:

Statement: All trinomials that have a perfect square in the first and last terms can be factored.

Consider any two perfect square terms,

Let the terms be $x^2,4$

Forming a trinomial with the first and last perfect terms need not be factored.

Consider,

  $x^2+10x+4$

Here, the first and last terms are perfect squares, but still this cannot be factored. Thus, all trinomials that have a perfect square in the first and last terms cannot be factored.

Conclusion:

The given statement is false.