/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 48 Write the function in the form \... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 48

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k,\) and demonstrate that \(f(k)=r\). \(f(x)=x^{3}-5 x^{2}-11 x+8, \quad k=-2\)

Short Answer

Expert verified
The original polynomial can be expressed as \(f(x)=(x+2)(x^{2}-7x+3)+14\), and \(f(-2) = 14\), which matches our remainder \(r\), confirming the correct division.

Step by step solution

01

Setup the synthetic division

To set up synthetic division, write the coefficients of \(f(x)\) in the top row, and place the value of \(k\) outside the division bar, on the left. So the setup is: \[\begin{{array}}{{r}}-2 \hspace{0.25in} | \hspace{0.25in} 1 \hspace{0.25in} -5 \hspace{0.25in} -11 \hspace{0.25in} 8 \\end{{array}}\]
02

Perform the synthetic division

First, bring down the first coefficient in the dividend, which is 1. Then for each next coefficient, multiply the previous result by the divisor \(-2\) and add it to the current coefficient.This gives us the coefficients for \(q(x)\) and the remainder \(r\):\[\begin{{array}}{{r}}-2 \hspace{0.25in} | \hspace{0.25in} 1 \hspace{0.25in} -5 \hspace{0.25in} -11 \hspace{0.25in} 8 \\hspace{0.5in} \hspace{0.25in} | \hspace{0.25in} -2 \hspace{0.25in} 14 \hspace{0.25in} 6 \\hspace{0.5in} \hspace{0.25in} | \hspace{0.25in} \underline{+}\hspace{0.25in} \underline{+}\hspace{0.25in} \underline{+} \\hspace{0.5in} \hspace{0.25in} | \hspace{0.25in} 1 \hspace{0.25in} -7 \hspace{0.25in} 3 \hspace{0.25in} 14 \end{{array}}\]
03

Write out \(q(x)\) and \(r\)

From the last row of our synthetic division, we can see that the coefficients for \(q(x)\) are 1, -7, and 3, and the remainder \(r\) is 14. So the polynomial \(q(x) = x^2 -7x +3\), and \(r = 14\). Thus the original function can be written as \(f(x)=(x+2)(x^{2}-7x+3)+14\).
04

Check our answer

To confirm that we have the correct expression for \(f(x)\), we check that \(f(-2) = 14\), which is our remainder \(r\). Substituting \(-2\) for \(x\) in the expanded expression \((x+2)(x^{2}-7x+3)+14\), we get: \[f(-2) = (-2+2)((-2)^{2}-7*(-2)+3)+14 = 0+14 = 14,\]which confirms that our division is correct.

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