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Chapter 3: Problem 75

When \(2 x^{2}-7 x+9\) is divided by a polynomial, the quotient is \(2 x-3\) and the remainder is \(3 .\) Find the polynomial.

Short Answer

Expert verified
The polynomial \(g(x)\) that, when dividing \(2 x^{2}-7 x+9\), yields the quotient \(2 x-3\) and the remainder \(3\) is \(g(x) = \frac{2 x^{2} -7 x + 6}{2 x - 3}\).

Step by step solution

01

Understand the division algorithm for polynomials

If \(p(x)\) is divided by \(g(x)\), then polynomials \(q(x)\) and \(r(x)\) will exist such that \(p(x) = g(x)q(x) + r(x)\) where either \(r(x)\) = 0, or degree[\(r(x)\)] < degree[\(g(x)\)]. Here, \(p(x)\) is the dividend, \(g(x)\) is the divisor, \(q(x)\) is the quotient and \(r(x)\) is the remainder.
02

Plug values into division algorithm equation

Here, we know that our dividend \(p(x)=2 x^{2}-7 x+9\), our quotient \(q(x)=2 x-3\), and remainder \(r(x)=3\). Therefore, the problem states that \(p(x) = g(x)q(x) + r(x)\) becomes \(2 x^{2}-7 x+9 = g(x)(2 x-3) + 3\). Now, we need to solve this equation for \(g(x)\).
03

Solve for divisor \(g(x)\)

To solve for \(g(x)\), we can rewrite the equation to isolate \(g(x)\) on one side. In this case, we subtract 3 from both sides and divide by \(2 x-3\), so \(g(x) = \frac{2 x^{2}-7 x+9 - 3}{2 x-3}\). Simplifying the numerator, we get \(g(x) = \frac{2 x^{2} -7 x + 6}{2 x - 3}\).

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Most popular questions from this chapter

Use point plotting to graph \(f(x)=2^{x}\). Begin by setting up a partial table of coordinates, selecting integers from \(-3\) to 3 inclusive, for \(x\). Because \(y=0\) is a horizontal asymptote, your graph should approach, but never touch, the negative portion of the \(x\) -axis.

Determine whether cach statement is true or false If bhe statement is false, make the necessary change(s) to produce a true statement. It is possible to have a rational function whose graph has no \(y\) -intercept.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, I only pay attention to the sign of \(f\) at each test value and not the actual function value.

Explain what is meant by combined variation. Give an example with your explanation.

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies directly as \(x . y=45\) when \(x=5 .\) Find \(y\) when \(x=13 .\)
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