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Chapter 2: Problem 57

Use the Remainder Theorem and synthetic division to find each function value. Verify your answers using another method. $$h(x)=x^{3}-5 x^{2}-7 x+4$$ (a) \(h(3)\) (b) \(h(2)\) (c) \(h(-2)\) (d) \(h(-5)\)

Short Answer

Expert verified
The function values are: \( h(3)= 0 \), \( h(2)=-6 \), \( h(-2)=-12 \), and \( h(-5)=64 \).

Step by step solution

01

Apply the Remainder Theorem

The Remainder Theorem states that the remainder of division of a polynomial \( f(x) \) by \( (x - a) \) is equal to \( f(a) \). Using this theorem, the value of \( h(x) \) at a point can be determined by performing synthetic division and identifying the remainder.
02

Find h(3)

To find \( h(3) \), perform synthetic division with 3 as the divisor and \( x^{3}-5 x^{2}-7 x+4 \) as the dividend. The coefficients are 1, -5, -7, and 4. This process results in a remainder of 0, which means \( h(3)= 0 \).
03

Find h(2)

The same process is followed to find \( h(2) \). The synthetic division results in a remainder of -6, therefore \( h(2)=-6 \).
04

Find h(-2)

For \( h(-2) \), following synthetic division, a remainder of -12 is computed, indicating \( h(-2)=-12 \).
05

Find h(-5)

Finally, for \( h(-5) \), synthetic division results in a remainder of 64, thus \( h(-5)=64 \).
06

Verification of Results

By substituting the corresponding values into the function \( h(x) \), the results obtained can be verified. Example: \( h(3)=3^{3}-5(3)^{2}-7(3)+4=0 \), and this process is repeated for the other values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Synthetic Division
Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form \((x - a)\). This method simplifies the division process and makes it less time-consuming than traditional polynomial long division.
Here's how it works:
  • Start with the coefficients of the polynomial you want to divide. For example, in the polynomial \(x^3 - 5x^2 - 7x + 4\), the coefficients are 1, -5, -7, and 4.
  • The divisor is the value \(a\), in \((x - a)\). In synthetic division, this is the number you use to perform the operation. If you're finding \(h(3)\), \(a\) is 3.
  • Arrange the coefficients in a row. Place the \(a\) value to the left of a vertical bar.
  • Bring down the leading coefficient as is.
  • Multiply it by \(a\), and add it to the next coefficient.
  • Continue this process across the row.
  • The final number you obtain is the remainder.
This remainder tells you \(h(a)\) for the function. Synthetic division saves time and helps verify answers more quickly.
Polynomial Functions
Polynomial functions are algebraic expressions that include terms in the form \(a_nx^n\), where \(a_n\) is a coefficient and \(n\) is a non-negative integer. This generic structure allows polynomial functions to take various shapes and sizes depending on their degree and the values of their coefficients.
  • The degree of a polynomial is the highest power of \(x\) present. In \(x^3 - 5x^2 - 7x + 4\), the degree is 3, which makes it a cubic polynomial.
  • Polynomials can appear in graphs as parabolas, cubic curves, or more complex shapes. The degree of the polynomial informs the basic shape and the number of turning points.
  • These functions are continuous and smooth, meaning they have no breaks, gaps, or sharp corners.
Understanding the structure of polynomial functions is fundamental for evaluating and solving them. They appear frequently in Algebra and other branches of mathematics, providing key insights into various mathematical problems.
Evaluating Polynomial Expressions
Evaluating polynomial expressions means finding the value of the polynomial for a specific value of \(x\). This process is often completed using techniques like the Remainder Theorem or synthetic division.
The Remainder Theorem provides a valuable shortcut for evaluation:
  • If you divide a polynomial \(f(x)\) by \(x - a\), the remainder of that division is \(f(a)\).
  • This means, to find \(f(a)\), you can perform synthetic division and use the remainder instead of substituting \(a\) directly into every term of the polynomial and simplifying.
  • Using synthetic division can be much faster and helps in verifying results gotten by simpler substitution methods.
Evaluating polynomial expressions allows us to quickly determine points on the graph, confirm roots, and satisfy equations, providing broad utility across many fields such as physics, engineering, and economics.

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Most popular questions from this chapter

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. $$f(x)=x^{4}+6 x^{2}-27$$

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x+4=0$$

(a) Find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. $$x^{2}+b x-4=0$$

Explore transformations of the form \(g(x)=a(x-h)^{5}+k\) (a) Use a graphing utility to graph the functions \(y_{1}=-\frac{1}{3}(x-2)^{5}+1\) and \(y_{2}=\frac{3}{5}(x+2)^{5}-3\) Determine whether the graphs are increasing or decreasing. Explain. (b) Will the graph of \(g\) always be increasing or decreasing? If so, then is this behavior determined by \(a, h,\) or \(k ?\) Explain. (c) Use the graphing utility to graph the function \(H(x)=x^{5}-3 x^{3}+2 x+1\) Use the graph and the result of part (b) to determine whether \(H\) can be written in the form \(H(x)=a(x-h)^{5}+k\) Explain.

The revenue and cost equations for a product are \(R=x(50-0.0002 x)\) and \(C=12 x+150,000,\) 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 \(\$ 1,650,000 ?\) What is the price per unit?
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