In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (x^2+8x+15)÷(x+5)

Verified step by step guidance

1. Step 1: Set up the long division by writing $x^2+8x+15$ under the division symbol and $x+5$ outside.
2. Step 2: Divide the first term of the dividend $x^2$ by the first term of the divisor $x$, which gives $x$.
3. Step 3: Multiply $x$ by the entire divisor $x+5$ to get $x^2+5x$, and write this product under the dividend.
4. Step 4: Subtract $x^2+5x$ from $x^2+8x+15$ to get a new polynomial $3x+15$.
5. Step 5: Repeat the process: divide $3x$ by $x$ to get $3$, multiply $3$ by $x+5$ to get $3x+15$, subtract to find the remainder, which 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, 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. According to the polynomial division algorithm, any polynomial can be expressed as the product of the divisor and the quotient, plus the remainder.

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors, which can simplify division and other operations. For example, the polynomial x^2 + 8x + 15 can be factored into (x + 3)(x + 5). Recognizing factors can help in both simplifying expressions and verifying the results of polynomial division, as the factors can provide insight into the roots and behavior of the polynomial.