Question

What are the zeroes of the function? What are their multiplicities? = x4 – 4x3 + 3x2

Answer 1

Answer-

Zeros of the functions are 0 with multiplicity of 2 and 1, 3 with multiplicity of 1.

Solution-

The given polynomial is,

$x^4-4x^3+3x^2$

For calculating roots,

$\Rightarrow x^4-4x^3+3x^2=0$

$\Rightarrow x^4-3x^3-x^3+3x^2=0$

$\Rightarrow x(x^3-3x^2)-1(x^3-3x^2)=0$

$\Rightarrow (x-1)(x^3-3x^2)=0$

$\Rightarrow (x-1)(x^2(x-3))=0$

$\Rightarrow (x^2)(x-1)(x-3)=0$

The root at x²=0, has multiplicity of 2 as the power is 2.

So, the graph will bounce at x=0

$\Rightarrow x^2=0,x-1=0,x-3=0$

$\Rightarrow x=0,1,3$

At x=1, x=3 the graph will go through the x axis. They have multiplicity of 1.

Answer 2

The zeroes of the function are $\boxed{0, 1 \text{and } 3}$. Multiplicity of $0$ is $\boxed{\bf 2}$, multiplicity of $1$ is $\boxed{\bf 1}$ and the multiplicity of $3$ is $\boxed{\bf 1}$.

Further explanation:

The given polynomial is $x^{4}-4x^{3}+3x^{2}$.

The above polynomial is a biquadratic polynomial.

Consider the given polynomial as follows:

$\boxed{P(x)=x^{4}-4x^{3}+3x^{2}}$  

The polynomial $P(x)$ is a polynomial of degree $4$.

To find the zeroes of polynomial $P(x)$ equate this polynomial to 0.

$\begin{aligned}P(x)&=0\\x^{4}-4x^{3}+3x^{2}&=0\end{aligned}$  

Factorize the polynomial $P(x)$ by taking common term $x^{2}$, since $x^{2}$ is the only lowest term in above polynomial with power $2$.

Factorize the polynomial $P(x)$ as follows:

$x^{2}(x^{2}-4x+3)=0$

Now, factorize the quadratic term $x^{2}-4x+3$ as follows:

$\begin{aligned}x^{2}-4x+3&=x^{2}-3x-x+3\\&=x(x-3)-1(x-3)\\&=(x-1)(x-3)\end{aligned}$

Substitute $(x-1)(x-3)$ in the polynomial $P(x)$ as shown below:

$x^{2}(x-1)(x-3)=0$

Now, from zero-product property the zeroes of the polynomial are shown below:

$\begin{aligned}x^{2}(x-1)(x-3)&=0\\x&=0,1,3\end{aligned}$

Here, $0$ is a zero of multiplicity $2$, since the power of $x$ is $2$.

Since, the degree of the polynomial is $4$, therefore, there will be $4$ solution to this polynomial.

Thus, the zeroes of the function $P(x)=x^{4}-4x^{3}+3x^{2}$ are $\bf 0, 1 \text{and } 3$ with $0$ having multiplicity of $\bf 2$ and both $1$ and $3$ have multiplicity of $\bf 1$.

Learn more:

1. Learn more about numbers brainly.com/question/4736384

2. Learn more about scientific notation of numbers brainly.com/question/4736384

Answer details:

Grade: High school

Subject: Mathematics

Chapter: Polynomial

Keywords: Zeroes, function, multiplicity, x^4-4x^3+3x^2, degree, highest power, polynomial, quadratic, equation, zero-product property, factorization, biquadratic expression.