Chapter 8, Problem 77SGR

To determine

To factorize the given polynomial

Answer to Problem 77SGR

  $49(x-\frac{2}{7})(x+\frac{2}{7})$

Explanation of Solution

Given:

Polynomial: $4-28a+49a^2$

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,

  $\Rightarrow 4-28a+49a^2$

Substituting in the above formula where $a=49,b=-28,c=4$

  $\begin{array}{l}\Rightarrow x=\frac{-(-28)\pm \sqrt{{(-28)}^2-4(49)(4)}}{2(49)}\\ \Rightarrow x=\frac{28\pm \sqrt{784-784}}{98}\\ \Rightarrow x=\frac{28\pm 0}{98}\\ \Rightarrow x=\frac{2\pm 0}{7}\\ \Rightarrow x=\frac{2}{7},\frac{2}{7}\end{array}$

The roots are,

  $\Rightarrow x_1=x_2=\frac{2}{7}$

The roots of the quadratic polynomial $4-28a+49a^2=49(x-\frac{2}{7})(x+\frac{2}{7}).$

Conclusion:

Therefore, the factors of the polynomial are $49(x-\frac{2}{7})(x+\frac{2}{7}).$