Question

select the correct location in the expression.

consider this expression.

\\\frac{4x^4 - 6x^3 + 6x + 3}{x - 3}\\

which term in the quotient of this expression contains an error?

\\4x^3 + 6x^2 + 18x + 78 + \frac{183}{x - 3}\\

Explanation:

Perform synthetic division of the polynomial expression

The given expression is:

$$ \frac{4x^4 - 6x^3 + 6x + 3}{x - 3} $$

Write the coefficients of the dividend \(4x^4 - 6x^3 + 0x^2 + 6x + 3\) and use \(c = 3\) for synthetic division:

$$
LATEXBLOCK0
$$

Compare the calculated quotient with the given expression

The correct quotient and remainder are:

$$ 4x^3 + 6x^2 + 18x + 60 + \frac{183}{x - 3} $$

The given quotient expression is:

$$ 4x^3 + 6x^2 + 18x + 78 + \frac{183}{x - 3} $$

Comparing the terms:

• \(4x^3\) is correct.
• \(6x^2\) is correct.
• \(18x\) is correct.
• \(78\) is incorrect (it should be \(60\)).
• \(\frac{183}{x - 3}\) is correct.

Identify the term containing the error

The term containing the error is \(78\).

Answer:

• \(4x^3\)
• \(6x^2\)
• \(18x\)
• \(78\) (Correct answer)
• \(\frac{183}{x - 3}\)