Q13. - Algebra 1

Question

Determine whether each trinomial is a perfect square trinomial. Write yes or no. If so, factor it.

$k^2-16k+64$

Step-by-Step Solution

Answer

Yes, the given trinomial is a perfect square trinomial.

The factorization of the given trinomial is ${\left(k-8\right)}^{\mathbf{2}}$.

Step 1. Observe the given trinomial k 2 − 16 k + 64 .

The given trinomial is: $k^2-16k+64$

The First, middle, and last terms of the given trinomial are $k^2,-16k$ and 64 respectively.

The first term of the given trinomial can be written as:

$k^2={\left(k\right)}^2$

Therefore, the first term of the given trinomial is a perfect square.

The last term of the given trinomial can be written as:

$64={\left(8\right)}^2$

Therefore, the last term of the given trinomial is a perfect square.

The middle term of the given trinomial can be written as:

 $-16k=-2\left(k\right)\left(8\right)$

Therefore, the middle term is twice the product of the square roots of the first term and last term.

Step 2. Determine whether the given trinomial k 2 − 16 k + 64 is a perfect square trinomial .

As, the first and last terms of the given trinomial are a perfect square and the middle term is twice the product of the square roots of the first term and last term.

Therefore, yes, the given trinomial is a perfect square trinomial.

Step 3. Factor the given trinomial k 2 − 16 k + 64 .

It is known that:

$a^2-2ab+b^2=\left(a-b\right)\left(a-b\right)={\left(a-b\right)}^2$

It can be noticed that:

$$\begin{gathered} k^2-16k+64={\left(k\right)}^2-2\left(k\right)\left(8\right)+{\left(8\right)}^2 \\ =\left(k-8\right)\left(k-8\right)\left(\because a^2-2ab+b^2=\left(a-b\right)\left(a-b\right)={\left(a-b\right)}^2\right) \\ ={\left(k-8\right)}^2 \end{gathered}$$

Therefore, the factorization of the given trinomial is ${\left(k-8\right)}^2$.