Find the zeros for each function 1. f(x)= x^3 + 12x^2 + 21x + 10

2. f(x) = x^4 + x^3 - 11x^2 - 9x + 18

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Step-by-step explanation:

For higher-degree polynomials I find it extremely useful to use a graphing calculator to find the zeros.

The usual approaches are ...

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1. Rational roots will be factors of 10, so ±1, ±2, ±5, or ±10. The signs are all positive, so there are no positive real roots. The sum of odd-degree coefficients is the same as the sum of even-degree coefficients, so -1 is one of the roots. Removing that using synthetic division gives you ...

   f(x) = (x +1)(x^2 +11 +10)

The quadratic (more or less obviously) factors as (x+1)(x+10), so the complete factorization is ...

  f(x) = (x +1)^2(x +10)

and the zeros are {-10, -1, -1}.

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2. Rational roots will be factors of 18, so will be ±1, ±2, ±3, ±6, ±9, ±18. The sum of coefficients is 0, so x=1 is a root. The rule of signs tells you that there are 2 positive real roots, so there is one more positive real root. Synthetic division tells you the factoring so far is ...

  f(x) = (x -1)(x^3 +2x^2 -9x -18)

We note that the first two coefficients have the same ratio as the last two, so we can factor by grouping.

  x^3 +2x^2 -9x -18 = x^2(x +2) -9(x +2) = (x^2 -9)(x +2)

Factoring the difference of squares gives you the complete factorization, hence all of the roots. (The zeros are the values of x that make the factors zero.)

  f(x) = (x -1)(x +2)(x -3)(x +3)

The zeros are {-3, -2, 1, 3}.

an advantage of the standard deviation is that it increases as the dispersion of the data increases. e. the interquartile range is preferred when the data are not skewed or no have outliers. an advantage of the standard deviation is that it uses all the observations in its computation.

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