Write a polynomial function in standard form with zeros of 0 (multiplicity 2), 1 (multiplicity 2), and 5/2 (multiplicity 2).
Understand the Problem
The problem asks us to construct a polynomial function given its zeros and their multiplicities. The zeros are 0 (multiplicity 2), 1 (multiplicity 2), and 5/2 (multiplicity 2). We need to write the polynomial in standard form, meaning expanded and simplified.
Answer
$f(x) = x^6 - 7x^5 + \frac{69}{4}x^4 - \frac{35}{2}x^3 + \frac{25}{4}x^2$
Answer for screen readers
$f(x) = x^6 - 7x^5 + \frac{69}{4}x^4 - \frac{35}{2}x^3 + \frac{25}{4}x^2$
Steps to Solve
- Write the polynomial in factored form
Given the zeros and their multiplicities, we can write the polynomial in factored form as: $$f(x) = a(x-0)^2(x-1)^2(x-\frac{5}{2})^2$$ Since the problem doesn't specify a leading coefficient, we can assume $a = 1$.
- Simplify the factored form
Simplify the expression: $$f(x) = x^2(x-1)^2(x-\frac{5}{2})^2$$
- Expand $(x-1)^2$
Expand the term $(x-1)^2$: $$(x-1)^2 = x^2 - 2x + 1$$
- Expand $(x - \frac{5}{2})^2$
Expand the term $(x - \frac{5}{2})^2$: $$(x - \frac{5}{2})^2 = x^2 - 5x + \frac{25}{4}$$
- Substitute the expanded terms back into the polynomial
Substitute the expanded terms into the polynomial: $$f(x) = x^2(x^2 - 2x + 1)(x^2 - 5x + \frac{25}{4})$$
- Multiply the quadratic terms
Multiply the two quadratic terms: $$(x^2 - 2x + 1)(x^2 - 5x + \frac{25}{4}) = x^4 - 5x^3 + \frac{25}{4}x^2 - 2x^3 + 10x^2 - \frac{25}{2}x + x^2 - 5x + \frac{25}{4}$$ Combine like terms: $$x^4 - 7x^3 + (\frac{25}{4} + 10 + 1)x^2 + (-\frac{25}{2} - 5)x + \frac{25}{4}$$ $$x^4 - 7x^3 + \frac{69}{4}x^2 - \frac{35}{2}x + \frac{25}{4}$$
- Multiply by $x^2$
Multiply the result by $x^2$: $$f(x) = x^2(x^4 - 7x^3 + \frac{69}{4}x^2 - \frac{35}{2}x + \frac{25}{4}) = x^6 - 7x^5 + \frac{69}{4}x^4 - \frac{35}{2}x^3 + \frac{25}{4}x^2$$
- Write the polynomial in standard form
The polynomial in standard form is: $$f(x) = x^6 - 7x^5 + \frac{69}{4}x^4 - \frac{35}{2}x^3 + \frac{25}{4}x^2$$
$f(x) = x^6 - 7x^5 + \frac{69}{4}x^4 - \frac{35}{2}x^3 + \frac{25}{4}x^2$
More Information
The polynomial function with the given zeros and multiplicities is $f(x) = x^6 - 7x^5 + \frac{69}{4}x^4 - \frac{35}{2}x^3 + \frac{25}{4}x^2$. The degree of the polynomial is 6, which is the sum of the multiplicities of the zeros (2 + 2 + 2 = 6).
Tips
A common mistake is not squaring the $(x - \frac{5}{2})$ term. Another common mistake is making errors when expanding and simplifying the polynomial. Remember to carefully distribute and combine like terms. Also, be careful when dealing with fractions.
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