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Chapter 5: Problem 106

Suppose that \(f(x)=4 x^{3}-11 x^{2}-26 x+24\). Find the zeros of \(f(x-2)\).

Short Answer

Expert verified
Solve \(4y^3 + 13y^2 - 22y - 40 = 0\) for \(y\), then substitute back \( x = y + 2 \).

Step by step solution

01

Understand the problem

The task is to find the zeros of the transformed function. First, recognize that the function given is a cubic polynomial: \(f(x) = 4x^3 - 11x^2 - 26x + 24\). We need to find the zeros of \(f(x-2)\).
02

Substitute \(x - 2\) into the function

To obtain \(f(x-2)\), substitute \(y = x - 2\) into \(f(y)\). First rewrite the function in terms of \(y\): \(f(y+2)\).
03

Substitute and simplify

Substitute \(y + 2\) into the original function: \[ f(y+2) = 4(y+2)^3 - 11(y+2)^2 - 26(y+2) + 24 \] Expand each term and simplify.
04

Expand and Combine like terms

Expand each polynomial term: \[ (y+2)^3 = y^3 + 6y^2 + 12y + 8 \] \[ (y+2)^2 = y^2 + 4y + 4 \] Substitute these expansions back into \(f(y+2)\):\[ f(y+2) = 4(y^3 + 6y^2 + 12y + 8) - 11(y^2 + 4y + 4) - 26(y+2) + 24 \] Combine like terms after distributing: \[ f(y+2) = 4y^3 + 24y^2 + 48y + 32 - 11y^2 - 44y - 44 - 26y - 52 + 24 \] Simplify: \[ f(y+2) = 4y^3 + 13y^2 - 22y - 40 \]
05

Find the zeros of the simplified function

Set \( f(y+2) = 0 \) and solve for \( y \): \[ 4y^3 + 13y^2 - 22y - 40 = 0 \] Use factorization techniques (e.g., synthetic division, trial and error with rational root theorem) to find the values of y that satisfy the equation.
06

Solve for x

Once the roots of \(f(y)\) are found, where \( y = x - 2 \), solve for \( x \) by adding 2 back to each root found for \( y \).

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Key Concepts

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

cubic polynomials
A cubic polynomial is a type of polynomial where the highest degree of the variable is three. For example, in the given function, we have, \(f(x) = 4x^3 - 11x^2 - 26x + 24\). Here:
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Most popular questions from this chapter

Use the Rational Zeros Theorem to find all the real zeros of each polynomial function. Use the zeros to factor \(f\) over the real numbers. $$ f(x)=x^{3}+8 x^{2}+11 x-20 $$

Solve each equation in the real number system. $$ 3 x^{3}+4 x^{2}-7 x+2=0 $$

Create a rational function with the following characteristics: three real zeros, one of multiplicity \(2 ; y\) -intercept 1 ; vertical asymptotes, \(x=-2\) and \(x=3 ;\) oblique asymptote, \(y=2 x+1\). Is this rational function unique? Compare your function with those of other students. What will be the same as everyone else's? Add some more characteristics, such as symmetry or naming the real zeros. How does this modify the rational function?

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