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Findong rational zeros and factoring polynomials

1) Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. If a zero has multiplicity greater than one, only enter the root once.)
P(x) = x^3 â?' 6x^2 + 11x â?' 6
x=

Write the polynomial in factored form.
P(x) =

2) Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. If a zero has multiplicity greater than one, only enter the root once.)
P(x) = x^3 + 9x^2 â?' 48x â?' 448
x=

Write the polynomial in factored form.
P(x) =

3)Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. If a zero has multiplicity greater than one, only enter the root once.)
P(x) = x^3 â?' x^2 â?' 65x + 225
x=

Write the polynomial in factored form.
P(x) =

4)Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. If a zero has multiplicity greater than one, only enter the root once.)
P(x) = x^4 â?' 13x^2 + 36
x=

Write the polynomial in factored form.
P(x) =

5)Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list. If a zero has multiplicity greater than one, only enter the root once.)
P(x) = x^4 â?' x^3 â?' 42x^2 â?' 32x + 224
x=

Write the polynomial in factored form.
P(x) =

Solution Preview

If a polynomial p(x) = a_n x^n + ... + a_1 x + a_0 has a rational zero p/q, then p is a factor of a_0 and q is a factor of a_n. In the leading coefficient a_n is equal to one, we just find the factors of the constant term and try them for being roots of our polynomial. Notice, that when we know at least one root, say, c, of our polynomial, we can then divide p(x) by (x-c) and get a polynomial of a lesser degree to work with.

1) P(x) = x^3 â?' 6x^2 + 11x â?' 6
The integer factors of 6 are -6, -3, -2, -1, 1, 2, 3 and 6. Trying them in the equation x^3-6x^2+11x -6=0, we ...

Solution Summary

The process of finding rational zeros and subsequent factoring a polynomial is explained and illustrated with several examples

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