Question

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try: take the first steps of polynomial long division
\\(\dfrac{6x^3 + 3x^2 - 14x - 2}{2x + 1}\\)

in the rational expression above, the numerator is a \\(\text{degree}\\) polynomial and the denominator is a first degree polynomial. therefore, the quotient must be a \\(\text{degree}\\) polynomial.

to find the first term of the quotient, we must divide the highest degree term in the numerator, \\(\square\\), by the highest degree term in the denominator, \\(2x\\).

the first term of the quotient is \\(\square\\).

check
explain

Explanation:

Step1: Determine the degree of the numerator polynomial

The numerator is \(6x^{3}+3x^{2}-14x - 2\). The highest power of \(x\) in this polynomial is \(3\), so the numerator is a third - degree polynomial.

Step2: Determine the degree of the quotient polynomial

When dividing a polynomial of degree \(n\) by a polynomial of degree \(m\) (\(n\geq m\)), the degree of the quotient polynomial is \(n - m\). Here, the numerator (dividend) is a third - degree polynomial (\(n = 3\)) and the denominator (divisor) is a first - degree polynomial (\(m=1\)). So the degree of the quotient polynomial is \(3 - 1=2\), so the quotient must be a second - degree polynomial.

Step3: Find the highest - degree term in the numerator

The highest - degree term in the numerator \(6x^{3}+3x^{2}-14x - 2\) is \(6x^{3}\).

Step4: Find the first term of the quotient

To find the first term of the quotient, we divide the highest - degree term of the numerator (\(6x^{3}\)) by the highest - degree term of the denominator (\(2x\)). Using the rule of exponents for division of variables with exponents (\(\frac{x^{a}}{x^{b}}=x^{a - b}\)) and the rule for division of coefficients (\(\frac{a}{b}\) where \(a\) and \(b\) are coefficients), we have \(\frac{6x^{3}}{2x}=\frac{6}{2}\times x^{3-1}=3x^{2}\).

Answer:

• The numerator is a third - degree polynomial.
• The quotient must be a second - degree polynomial.
• The highest - degree term in the numerator is \(6x^{3}\).
• The first term of the quotient is \(3x^{2}\).