Mathematics, 03.02.2020 14:03 kaliloabousjbf

What are the zeroes of the function? what are their multiplicities?

= x4 – 4x3 + 3x2

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Answer from: live4dramaoy0yf9

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.

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

Answer from: soliseric879

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

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 .

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, is a zero of multiplicity , since the power of is .

Since, the degree of the polynomial is , therefore, there will be 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 having multiplicity of $\bf 2$ and both and have multiplicity of $\bf 1$ .

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2. Learn more about scientific notation of numbers link

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.

Answer from: Quest

step-by-step explanation:

Answer from: Quest

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