Perform each division. See Examples 9 and 10. (p²+2p+20)/(p+6)

Verified step by step guidance

1. Identify the division problem as a polynomial division: divide the polynomial \(p^{2} + 2p + 20\) by the binomial \(p + 6\).
2. Set up the long division by writing \(p^{2} + 2p + 20\) under the division symbol and \(p + 6\) outside.
3. Divide the leading term of the dividend \(p^{2}\) by the leading term of the divisor \(p\) to get the first term of the quotient: \(p\).
4. Multiply the entire divisor \(p + 6\) by this term \(p\) and subtract the result from the dividend to find the new remainder.
5. Repeat the process with the new remainder: divide its leading term by \(p\), multiply the divisor by this term, subtract again, and continue until the degree of the remainder is less than the degree of the divisor.

Key Concepts

Polynomial Division

Polynomial division is a process similar to long division with numbers, used to divide one polynomial by another. It helps simplify expressions or find quotients and remainders when dividing polynomials, especially when the divisor is a linear polynomial.

Long Division Method for Polynomials

The long division method involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by this result, subtracting from the dividend, and repeating with the remainder. This step-by-step approach breaks down complex polynomial division into manageable parts.

Remainder and Quotient in Polynomial Division

When dividing polynomials, the result includes a quotient and possibly a remainder. The quotient is the polynomial result of the division, while the remainder is a polynomial of lower degree than the divisor. Understanding these helps interpret the division outcome correctly.