Chapter 8, Problem 12SGR

To determine

To classify the polynomial as prime polynomial, difference of squares or perfect square trinomial.

Answer to Problem 12SGR

Perfect square trinomial

Explanation of Solution

Given:

Polynomial: $16x^2+40x+25$

Concept used:

Prime polynomial:

A polynomial with integer coefficients that cannot be reduced to a polynomial of a lower degree is a prime polynomial

Difference of squares:

A polynomial is called difference of squares when each term is a perfect square.

Difference of squares is of the form $a^2-b^2=(a+b)(a-b)$

Perfect square trinomial:

A perfect square trinomial is a polynomial with three terms which can be created by multiplying a binomial to itself.

Perfect square trinomial is of the form ${(a+b)}^2=a^2+2ab+b^2$

Formula used:

Roots of the polynomial of the form $a{x}^2+bx+c=a(x-x_1)(x-x_2)$ are

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation:

Consider the given polynomial,

  $\Rightarrow 16x^2+40x+25$

Solving the above polynomial using above formula,

  $\begin{array}{l}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-40\pm \sqrt{{40}^2-4(16)(25)}}{2(16)}\\ x=\frac{-40\pm \sqrt{1600-1600}}{32}\\ x=\frac{-40\pm 0}{32}\\ x=\frac{-5\pm 0}{4}\\ x=\frac{-5}{4},\frac{-5}{4}\end{array}$

Therefore, the equation can be written as

  $\begin{array}{l}\Rightarrow (x+\frac{5}{4})(x+\frac{5}{4})\\ \Rightarrow (4x+5)(4x+5)\\ $ \Rightarrow {(4x+5)}^2$\end{array}$

The equation can be written as $16x^2+40x+25={(4x+5)}^2.$

Conclusion:

Thus, the given polynomial is classified as perfect square trinomial.