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Chapter 3: Problem 53

How can the Factor Theorem be used to determine if \(x-1\) is a factor of \(x^{3}-2 x^{2}-11 x+12 ?\)

Short Answer

Expert verified
Yes, according to the Factor Theorem, \(x-1\) is a factor of the polynomial \(x^{3}-2x^{2}-11x+12\) because when \(x=1\) is substituted into the polynomial, the result equals to zero.

Step by step solution

01

Identifying the Value to Substitute into the Polynomial

The given factor to test is \(x-1\). To find the value of 'x' that must be used, set \(x-1 = 0\), which gives \(x=1\). Hence, \(x=1\) is substituted in the polynomial.
02

Substitution

Substitute \(x=1\) into the polynomial \(x^{3}-2x^{2}-11x+12\). It becomes \(1^{3} - 2(1^{2}) - 11(1) + 12\).
03

Simplifying the Polynomial

Simplify the polynomial. The result becomes \(1 - 2 - 11 + 12\). When simplified further, it equals to zero.
04

Interpreting the Result

Since the simplified result equals zero, it proves that under the Factor Theorem, \(x-1\) is a factor of the polynomial \(x^{3}-2x^{2}-11x+12\).

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Most popular questions from this chapter

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-2(x-4)^{2}\left(x^{2}-25\right)$$

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Write the equation of a rational function \(f(x)=\frac{p(x)}{q(x)}\) having the indicated properties, in which the degrees of \(p\) and \(q\) are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. \(f\) has a vertical asymptote given by \(x=1,\) a slant Esymptote whose equation is \(y=x, y\) -intercept at \(2,\) and \(x\) -intercepts at \(-1\) and 2.
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