Divide using long division. State the quotient, and the remainder, r(x). (x³+5x²+7x+2)÷(x+2)

Verified step by step guidance

1. Step 1: Set up the long division by writing the dividend \(x^3 + 5x^2 + 7x + 2\) under the division bar and the divisor \(x + 2\) outside the division bar.
2. Step 2: Divide the leading term of the dividend \(x^3\) by the leading term of the divisor \(x\). This gives the first term of the quotient, \(x^2\). Write \(x^2\) above the division bar.
3. Step 3: Multiply \(x^2\) by the divisor \(x + 2\), which results in \(x^3 + 2x^2\). Subtract \(x^3 + 2x^2\) from the dividend \(x^3 + 5x^2 + 7x + 2\), leaving \(3x^2 + 7x + 2\).
4. Step 4: Repeat the process with the new dividend \(3x^2 + 7x + 2\). Divide the leading term \(3x^2\) by \(x\), which gives \(3x\). Write \(3x\) in the quotient. Multiply \(3x\) by \(x + 2\), resulting in \(3x^2 + 6x\). Subtract \(3x^2 + 6x\) from \(3x^2 + 7x + 2\), leaving \(x + 2\).
5. Step 5: Divide the leading term \(x\) by \(x\), which gives \(1\). Write \(1\) in the quotient. Multiply \(1\) by \(x + 2\), resulting in \(x + 2\). Subtract \(x + 2\) from \(x + 2\), leaving a remainder of \(0\). The quotient is \(x^2 + 3x + 1\) and the remainder is \(0\).

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide polynomials, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the entire divisor by this result, and subtracting it from the dividend. This process is repeated with the new polynomial until the degree of the remainder is less than the degree of the divisor.

Quotient and Remainder

In polynomial division, the quotient is the result of the division, representing how many times the divisor fits into the dividend. The remainder is what is left over after the division process, which cannot be divided by the divisor anymore. The relationship can be expressed as: Dividend = Divisor × Quotient + Remainder.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It determines the polynomial's behavior and the number of roots it can have. Understanding the degree is crucial in polynomial division, as it helps identify when to stop the division process, specifically when the degree of the remainder is less than that of the divisor.