(b)
To find the quotient of x³-8 divided by x-2, we can use polynomial long division or synthetic division. However, we can also recognize that x³-8 is a difference of cubes. The difference of cubes formula is a³ - b³ = (a - b)(a² + ab + b²). Applying this formula, we set a = x and b = 2, thus:
x³ - 8 = x³ - 2³ = (x - 2)(x² + 2x + 4)
When we divide x³-8 by x-2, we can clearly see that the quotient is x² + 2x + 4, since x-2 divides evenly into x³-8 without any remainder.
Quotient: x² + 2x + 4
(c)
The division of one polynomial by another polynomial does not always result in another polynomial. It depends on the degrees of the polynomials involved. If we divide the polynomial x³ + 27 by x², we are trying to determine how many times x² goes into each term of the dividend.
x³ divided by x² gives us x (since x³ / x² = x¹).
27 divided by x² cannot be performed inside the realm of polynomials, as it would result in a fraction with a variable in the denominator (27 / x²).
Thus, the "quotient" would be x with a remainder of 27, but because we cannot express 27 / x² as a polynomial, the division does not result in another polynomial in this case.
Quotient: x (not a complete polynomial)
Remainder: 27 (not a polynomial)
(d)
When a polynomial of degree n is divided by a linear polynomial (which is a polynomial of degree 1), the quotient will generally be a polynomial of degree n - 1.
Explanation: When dividing by a linear polynomial, each term of the polynomial of degree n will be reduced by one degree. The highest term of the dividend (which is a_n*x^n) will become the highest term of the quotient (which will be something similar to a_n*x^(n-1) after dividing by a linear term like b_1*x). All lower degree terms in the original polynomial of degree n will also be reduced by one degree when forming the quotient.
Therefore, the degree of the quotient when a polynomial of degree n is divided by a linear polynomial will usually be n - 1, provided that the division is exact (meaning without a remainder that has the same degree as the original polynomial or higher).
Degree of Quotient: n - 1
