Perform each division. See Examples 9 and 10. (4x³+9x²-10x-6)/(4x+1)

Verified step by step guidance

1. Identify the dividend and divisor: the dividend is \$4x^3 + 9x^2 - 10x - 6\( and the divisor is \)4x + 1$.
2. Set up polynomial long division by writing \$4x + 1\( outside the division symbol and \)4x^3 + 9x^2 - 10x - 6$ inside.
3. Divide the leading term of the dividend (\$4x^3\() by the leading term of the divisor (\)4x\() to find the first term of the quotient: \)\frac{4x^3}{4x} = x^2$.
4. Multiply the entire divisor \$4x + 1\( by \)x^2$ and subtract the result from the dividend to find the new polynomial to divide.
5. Repeat the process: divide the new leading term by \$4x$, multiply the divisor by this term, subtract, and continue until the degree of the remainder is less than the degree of the divisor.

Key Concepts

Polynomial Long Division

Polynomial long division is a method used to divide a polynomial by another polynomial of lower degree, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying, subtracting, and repeating until the remainder has a lower degree than the divisor.

Leading Terms and Degree of Polynomials

The leading term of a polynomial is the term with the highest exponent, and the degree is the highest power of the variable. Understanding these helps determine the steps in division, as the division process focuses on reducing the degree of the dividend by matching leading terms with the divisor.

Remainder and Quotient in Polynomial Division

When dividing polynomials, the quotient is the result of the division, and the remainder is what is left when the division cannot continue because the degree of the remainder is less than the divisor. The division can be expressed as Dividend = Divisor × Quotient + Remainder.