1If \( f(x) = x^3 + x^2 + x + 1 \) and \( g(x) = x + 1 \), what is the degree of the quotient \( f(x) \div g(x) \)? Choose from 1, 2, 3 and 4.
2When a polynomial \( f(x) \) is divided by \( d(x) \), the quotient is \( q(x) \) and the remainder is \( R \). Which relation is correct: \( f(x) = d(x) \times q(x) + R \), \( f(x) = q(x) \times q(x) + R \), \( f(x) = R \times q(x) + d(x) \) or \( f(x) = d(x) \times R + R \)?
3What is the remainder when a polynomial \( f(x) \) is divided by \( x - a \)? Choose from 0, 1, \( f(x) \) and \( f(a) \).
4What is the remainder when a polynomial \( f(x) \) is divided by \( ax + b \)? Choose from \( f\left(\frac{a}{b}\right) \), \( f\left(-\frac{a}{b}\right) \), \( f\left(-\frac{b}{a}\right) \) and \( f\left(\frac{b}{a}\right) \).
5What is the remainder when \( x^3 - x^2 + 1 \) is divided by \( x - 1 \)? Choose from 0, 1, 2 and 3.
6When \( x^3 + 3x^2 - kx + 4 \) is divided by \( x - 2 \), the remainder is 4. What is the value of \( k \)? Choose from 10, 8, 4 and 2.
7State and prove the remainder theorem.
8Using the remainder theorem, find the remainder of \( (x^3 - x^2 + 1) \div (x - 2) \).
9Using the remainder theorem, find the remainder of \( (x^3 + 5x^2 - 6x - 16) \div (x + 1) \).
10Using the remainder theorem, find the remainder of \( (6x^3 + 4x^2 + 3x + 4) \div (3x - 4) \).
11Using the remainder theorem, find the remainder of \( (8x^4 + 5x^3 - 2x^2 + 7x - 1) \div (2x + 6) \).
12When \( f(x) = 2x^3 + 3x^2 - kx + 4 \) is divided by \( x + 2 \), the remainder is 16. Find the value of \( k \).
13When \( f(x) = 8x^3 + 4x^2 + (k - 3)x - 7 \) is divided by \( 2x - 1 \), the remainder is \( -2 \). Find the value of \( k \).
14When \( f(x) = x^3 - (p - 2)x^2 - px + 28 \) is divided by \( x + 3 \), the remainder is 10. Using the remainder theorem, find the value of \( p \).
15The polynomials \( f(x) = 2x^3 - (k - 2)x^2 - 7x + 14 \) and \( g(x) = 3x^3 + 8x^2 - kx + 6 \) leave equal remainders when each is divided by \( x + 2 \). Using the remainder theorem, find the value of \( k \).
16The polynomials \( f(x) = 2x^3 + hx^2 - 10x + 8 \) and \( g(x) = 6x^3 - hx^2 + 6 \) leave equal remainders when each is divided by \( 2x - 1 \). Using the remainder theorem, find the value of \( h \).
17When \( f(x) = x^3 + mx^2 - nx + 6 \) is divided by \( x + 2 \), the remainder is 24, and when it is divided by \( x - 3 \), the remainder is 54. Using the remainder theorem, find the values of \( m \) and \( n \).
18Divide \( x^3 + 2x^2 + 3x - 6 \) by \( x - 2 \) using long division. Then check the remainder with the remainder theorem.
19Find the remainder when \( x^3 - 6x^2 + 11x - 6 \) is divided by \( x - 3 \). What does the answer tell you about the division?
20Why is the remainder always a number, with no \( x \) in it, when a polynomial is divided by a linear divisor?
21To find the remainder when \( f(x) \) is divided by \( 3x + 2 \), which value of \( x \) should be substituted, and why?