Find the zeros of the function and state the multiplicities.

If there is more than one answer, separate them with commas. Select “None” if applicable.
Part:
Part 1 of 2
The zero(s) of :

Ask by Simmons Mullins. in the United States

Mar 20,2025

Question

Find the zeros of the function and state the multiplicities.

If there is more than one answer, separate them with commas. Select “None” if applicable.
Part:
Part 1 of 2
The zero(s) of :

Ask by Simmons Mullins. in the United States

Mar 20,2025

UpStudy AI · Accepted Answer

The polynomial is  
$$
k(x)=16x^{4}-17x^{2}+1.
$$
Notice that it is quadratic in $$x^2$$. Let  
$$
y=x^2.
$$
Then the equation becomes  
$$
16y^{2}-17y+1=0.
$$

We solve for $$y$$ using the quadratic formula:  
$$
y=\frac{17\pm\sqrt{17^2-4\cdot16\cdot1}}{2\cdot16}=\frac{17\pm\sqrt{289-64}}{32}=\frac{17\pm\sqrt{225}}{32}=\frac{17\pm15}{32}.
$$

The two solutions for $$y$$ are:  
$$
y=\frac{17+15}{32}=1 \quad \text{and} \quad y=\frac{17-15}{32}=\frac{1}{16}.
$$

Now substituting back $$x^2=y$$:

1. For $$y=1$$:  
   $$
   x^2=1 \quad\Longrightarrow\quad x=\pm 1.
   $$

2. For $$y=\frac{1}{16}$$:  
   $$
   x^2=\frac{1}{16} \quad\Longrightarrow\quad x=\pm \frac{1}{4}.
   $$

Thus, the zeros of $$ k(x) $$ are  
$$
-1, \; -\frac{1}{4}, \; \frac{1}{4}, \; 1.
$$

Each zero has multiplicity 1.
The zeros of $$ k(x) $$ are $$ -1, -\frac{1}{4}, \frac{1}{4}, $$ and $$ 1 $$, each with multiplicity 1.

Find the zeros of the function and state the multiplicities.

If there is more than one answer, separate them with commas. Select “None” if applicable.
Part:
Part 1 of 2
The zero(s) of :

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions

Find the zeros of the function and state the multiplicities. If there is more than one answer, separate them with commas. Select “None” if applicable. Part: Part 1 of 2 The zero(s) of :

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions

Find the zeros of the function and state the multiplicities.

If there is more than one answer, separate them with commas. Select “None” if applicable.
Part:
Part 1 of 2
The zero(s) of :