Question

What is the first term of the quotient when \(2x^3 + x^2 - 3x + 5\) is divided by \(1 - 3x + x^2\)?

Hint:

In polynomial division, the first term of the quotient is always found by dividing the highest degree term of the dividend by the highest degree term of the divisor.

Answer 1

Step 1: Rewrite the divisor in standard form.}
The divisor is given as \(1 - 3x + x^2\). Writing it in descending powers of \(x\), we get:
\[ x^2 - 3x + 1 \]
Step 2: Identify the leading terms.}
The leading term of the dividend \(2x^3 + x^2 - 3x + 5\) is \(2x^3\).
The leading term of the divisor \(x^2 - 3x + 1\) is \(x^2\).

Step 3: Divide the leading terms.}
To find the first term of the quotient, divide the leading term of the dividend by the leading term of the divisor:
\[ \frac{2x^3}{x^2} = 2x \]
Step 4: State the first term of the quotient.}
Hence, the first term of the quotient is:
\[ 2x \]