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Chapter 2: Problem 28

Find a polynomial \(p\) of degree 3 such that \(-2,-1,\) and 4 are zeros of \(p\) and \(p(1)=2\).

Short Answer

Expert verified
The polynomial is \(p(x) = -\frac{1}{9}(x+2)(x+1)(x-4)\). It has a degree of 3, zeros at -2, -1, and 4, and evaluates to 2 when x = 1, satisfying all the given conditions.

Step by step solution

01

(Factor Theorem)

We know the Factor Theorem states that if r is a zero of a polynomial, then the polynomial can be written as \(p(x) = (x-r)q(x)\), where q(x) is a polynomial. Since -2, -1, and 4 are all zeros of the polynomial, we can write \(p(x)\) as: \(p(x) = a(x+2)(x+1)(x-4)\)
02

(Substituting the given zeros and finding the value of 'a')

We are given that p(1) = 2. Let's substitute x = 1 into the polynomial. \(2 = a(1+2)(1+1)(1-4)\) Let's simplify it to obtain a. \(2 = a(3)(2)(-3)\) \(2 = -18a\) So, \(a = -\frac{1}{9}\)
03

(Construct Polynomial)

Now we can construct p(x) using the factorized form and the value of 'a' we just found. \(p(x) = -\frac{1}{9}(x+2)(x+1)(x-4)\) So, the polynomial is p(x) = \(-\frac{1}{9}(x+2)(x+1)(x-4)\). This polynomial has a degree of 3, zeros at -2, -1, and 4, and evaluates to 2 when x = 1, satisfying all the given conditions.

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Most popular questions from this chapter

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{x^{2}}{4 x+3} $$

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (r(x))^{2} $$

Suppose \(q(x)=2 x^{3}-3 x+1\) (a) Show that the point (2,11) is on the graph of \(q\). (b) Show that the slope of a line containing (2,11) and a point on the graph of \(q\) very close to (2,11) is approximately 21 . [Hint: Use the result of Exercise \(17 .]\)

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{4 x-5}{x+7} $$

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (s(x))^{2} $$
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