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Chapter 3: Problem 16

Use the factor theorem to show that \(x-c\) is a factor of \(f(x)\). $$f(x)=x^{4}-3 x^{3}-2 x^{2}+5 x+6 ; \quad c=2$$

Short Answer

Expert verified
Yes, \(x-2\) is a factor of \(f(x)\) because \(f(2) = 0\).

Step by step solution

01

State the Factor Theorem

The Factor Theorem states that \(x-c\) is a factor of a polynomial \(f(x)\) if and only if \(f(c) = 0\). In this exercise, we need to find whether \(x-2\) is a factor of the given polynomial \(f(x)\) by evaluating \(f(2)\).
02

Substitute and Evaluate f(c)

Substitute \(c = 2\) into the polynomial \(f(x) = x^4 - 3x^3 - 2x^2 + 5x + 6\). Calculate \(f(2)\):\[f(2) = (2)^4 - 3(2)^3 - 2(2)^2 + 5(2) + 6\]
03

Calculate Each Term

Break down and calculate each term separately:- \((2)^4 = 16\)- \(-3(2)^3 = -24\)- \(-2(2)^2 = -8\)- \(5(2) = 10\)- \(+6 = 6\)
04

Sum the Results

Add all the calculated terms together to find \(f(2)\):\[f(2) = 16 - 24 - 8 + 10 + 6\]
05

Simplify the Expression

Simplify the expression by performing the arithmetic:\[f(2) = (16 - 24) + (10 - 8) + 6 = -8 + 2 + 6 = 0\]
06

Conclusion

Since \(f(2) = 0\), according to the Factor Theorem, \(x-2\) is a factor of \(f(x)\).

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Key Concepts

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

Polynomial
A polynomial is an algebraic expression made up of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents. They serve as fundamental building blocks in mathematics, representing a vast range of problems and scenarios.
Examples of polynomials include:
  • Quadratic polynomials like \(ax^2 + bx + c\)
  • Cubic polynomials such as \(ax^3 + bx^2 + cx + d\)
  • Higher-degree polynomials like \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\)
A key feature of polynomials is their degree, which is the highest power of the variable. For instance, in the polynomial \(f(x) = x^4 - 3x^3 - 2x^2 + 5x + 6\), the degree is 4 because the highest power of \(x\) is 4. Polynomials are used to model a wide variety of scientific and engineering problems and serve as the foundation for calculus and algebra.
Polynomial Roots
The roots of a polynomial are the values of \(x\) which make the polynomial equal to zero. These are also known as "zeros" or "solutions." Finding the roots of a polynomial is essentially solving the equation \(f(x) = 0\).
Identifying the roots of a polynomial is important because:
  • They indicate where the graph of the polynomial crosses the x-axis.
  • Roots are essential in understanding the behavior of the polynomial function.
  • They are critical in engineering and physics for solving real-world problems.
In the context of the Factor Theorem, if \(f(c) = 0\), then \(x - c\) is a factor of the polynomial. For the polynomial \(f(x) = x^4 - 3x^3 - 2x^2 + 5x + 6\), when we substituted \(c = 2\), we found \(f(2) = 0\), confirming that \(x=2\) is a root, and thus \(x-2\) is a factor.
Synthetic Division
Synthetic division is a simplified method of dividing a polynomial by a linear divisor of the form \(x-c\). It is faster and more efficient than traditional long division when working with polynomials and is often used when applying the Factor Theorem.
The process of synthetic division involves:
  • Writing down the coefficients of the polynomial.
  • Using the root \(c\) as a divisor.
  • Performing a sequence of multiplications and additions to find the quotient and remainder.
For instance, to verify if \(x-2\) is a factor of \(f(x) = x^4 - 3x^3 - 2x^2 + 5x + 6\), you could use synthetic division. If it results in a remainder of 0, \(x-2\) is a factor. This method complements the Factor Theorem by providing a systematic approach to revealing polynomial factors efficiently.

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Most popular questions from this chapter

Constructing a box From a rectangular piece of cardboard having dimensions 20 inches \(\times 30\) inches, an open box is to be made by removing squares of area \(x^{2}\) from each comer and turning up the sides. (a) Show that there are two boxes that have a volume of 1000 in \(^{3}.\) (b) Which box has the smaller surface area?

Density at a point \(A\) thin flat plate is situated in an \(x y\) -plane such that the density \(d\) (in Ib/ft \(^{2}\) ) at the point \(P(x, y)\) is inversely proportional to the square of the distance from the origin. What is the effect on the density at \(P\) if the \(x\) - and \(y\) -coordinates are each multiplied by \(\frac{1}{3} ?\)

The average monthly temperatures in "F for two Canadian locations are Iisted in the following tables. $$\begin{array}{|l|llll|} \hline \text { Month } & \text { Jan. } & \text { Feb. } & \text { Mar. } & \text { Apr. } \\ \hline \text { Arctic Bay } & -22 & -26 & -18 & -4 \\ \hline \text { Trout Lake } & -11 & -6 & 7 & 25 \\ \hline \end{array}$$ $$\begin{array}{|l|cccc|} \hline \text { Month } & \text { May } & \text { June } & \text { July } & \text { Aug. } \\ \hline \text { Arctic Bay } & 19 & 36 & 43 & 41 \\ \hline \text { Trout Lake } & 39 & 52 & 61 & 59 \\ \hline \end{array}$$ $$\begin{array}{|l|cccc|} \hline \text { Month } & \text { Sept. } & \text { Oct. } & \text { Nov. } & \text { Dec. } \\ \hline \text { Arctic Bay } & 28 & 12 & -8 & -17 \\ \hline \text { Trout Lake } & 48 & 34 & 16 & -4 \\ \hline \end{array}$$ (a)If January 15 corresponds to \(x=1\), February 15 to \(x=2, \ldots,\) and December 15 to \(x=12,\) determine graphically which of the three polynomials given best models the data. (b)Use the Intermediate value theorem for polynomial functions to approximate an interval for \(x\) when an average temperature of \(0^{\circ} \mathrm{F}\) occurs. (c)Use your choice from part (a) to estimate \(x\) when the average temperature is \(0^{\circ} \mathrm{F}\). \(f(x)=-2.14 x^{2}+28.01 x-55\) \(g(x)=-0.22 x^{3}+1.84 x^{2}+11.70 x-29.90\) \(h(x)=0.046 x^{4}-1.39 x^{3}+11.81 x^{2}-22.2 x+1.03\)

Is there a polynomial of the given degree \(n\) whose graph contains the indicated points? $$\begin{aligned} &n=3\\\ &(1.1,-49.815),(2,0),(3.5,25.245),(5.2,0)\\\ &(6.4,-29.304),(10.1,0) \end{aligned}$$

\( Graph f,\) and find equations of the vertical asymptotes. $$f(x)=\frac{20 x^{2}+80 x+72}{10 x^{2}+40 x+41}$$
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