Divide using long division. State the quotient, and the remainder, r(x). (x²+8x+15)÷(x+5)

Verified step by step guidance

1. Step 1: Set up the long division. Write the dividend \(x^2 + 8x + 15\) under the long division symbol and the divisor \(x + 5\) outside the symbol.
2. Step 2: Divide the first term of the dividend \(x^2\) by the first term of the divisor \(x\). This gives \(x\). Write \(x\) as the first term of the quotient.
3. Step 3: Multiply \(x\) (the first term of the quotient) by the divisor \(x + 5\). This gives \(x^2 + 5x\). Subtract \(x^2 + 5x\) from \(x^2 + 8x + 15\), which results in \(3x + 15\).
4. Step 4: Divide the first term of the new dividend \(3x\) by the first term of the divisor \(x\). This gives \(3\). Write \(3\) as the next term of the quotient.
5. Step 5: Multiply \(3\) (the next term of the quotient) by the divisor \(x + 5\). This gives \(3x + 15\). Subtract \(3x + 15\) from \(3x + 15\), which results in \(0\). The quotient is \(x + 3\) and the remainder \(r(x)\) is \(0\).

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree. It involves a process similar to numerical long division, where you divide the leading term of the dividend by the leading term of the divisor, multiply the entire divisor by this result, and subtract it from the dividend. This process is repeated 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, while the remainder is what is left over after the division process. The relationship can be expressed as: Dividend = Divisor × Quotient + Remainder. Understanding how to identify and express both the quotient and the remainder is crucial for accurately completing polynomial long division.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors, which can simplify division problems. In the given exercise, recognizing that the polynomial x^2 + 8x + 15 can be factored into (x + 3)(x + 5) can provide insight into the division process. This understanding can also help verify the results of the long division by checking if the product of the quotient and divisor plus the remainder equals the original polynomial.