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Chapter 2: Problem 64

Write the polynomial as the product of linear factors and list all the zeros of the function. $$f(x)=x^{2}-x+56$$

Short Answer

Expert verified
The polynomial \(f(x)=x^{2}-x-56\) as the product of linear factors is \(f(x) = (x-8)(x+7)\). The zeros of the function are \(x = 8\) and \(x = -7\).

Step by step solution

01

Identify the Coefficients

The given function is \(f(x)=x^{2}-x-56\). Identifying the coefficients, \(a = 1\), \(b = -1\), and \(c = -56\).
02

Calculate the Discriminant

The discriminant of a quadratic equation is given by \(D = b^{2}-4ac\). Substituting the identified coefficients into this formula, we get \(D = (-1)^{2}-4(1)(-56)=1+224=225\).
03

Solve for the Zeros

The zeros of the function can be calculated using the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\). For our equation, this simplifies to \(x = \frac{1 \pm 15}{2} =\frac{16}{2}\) and \(\frac{-14}{2}\). So, \(x = 8\) and \(x = -7\) are the zeros.
04

Write the Polynomial as the Product of Linear Factors

Knowing the zeros of the function, it can be rewritten as the product of linear factors. Because the x-intercepts are 8 and -7, this can be expressed as \(f(x) = (x-8)(x+7)\).

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Most popular questions from this chapter

Prove that the complex conjugate of the sum of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the sum of their complex conjugates.

Find all the zeros of the function. When there is an extended list of possible rational zeros, use a graphing utility to graph the function in order to disregard any of the possible rational zeros that are obviously not zeros of the function. $$f(x)=x^{3}+24 x^{2}+214 x+740$$

The revenue and cost equations for a product are \(R=x(75-0.0005 x)\) and \(C=30 x+250,000,\) where \(R\) and \(C\) are measured in dollars and \(x\) represents the number of units sold. How many units must be sold to obtain a profit of at least \(\$ 750,000 ?\) What is the price per unit?

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. \(f(x)=x^{4}-3 x^{3}-x^{2}-12 x-20\) (Hint: One factor is \(\left.x^{2}+4 .\right)\)

Find all real zeros of the function. $$f(z)=12 z^{3}-4 z^{2}-27 z+9$$
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