1For \( f(x) = x^2 - 5x + 6 \), find \( f(2) \) and \( f(3) \). For which values of x is f(x) zero? Do \( x - 2 \) and \( x - 3 \) divide \( x^2 - 5x + 6 \) exactly?
2Divide \( f(x) = x^3 + 7x^2 - 12x + 4 \) by \( d(x) = x - 1 \). Find the quotient q(x) and the remainder R, and state the relation between f(x), d(x), q(x) and R.
3If \( (x - a) \) is a factor of a polynomial f(x), what is the value of \( f(a) \) when f(x) is divided by \( x - a \)? Choose from 0, 1, a and \( -a \).
4When a polynomial f(x) is divided by a linear factor, the remainder is \( f(-k) \). Which is that factor: \( x - a \), \( x + k \), \( x - k \) or \( x + a \)?
5Under which condition is \( (cx - d) \) a factor of the polynomial f(x)? Choose from \( f\left(\frac{c}{d}\right) = 0 \), \( f\left(-\frac{c}{d}\right) = 0 \), \( f\left(\frac{d}{c}\right) = 0 \) and \( f\left(-\frac{d}{c}\right) = 0 \).
6If \( (x - 1) \) is a factor of the polynomial \( x^3 + x^2 + p + 1 \), what is the value of p? Choose from \( -5 \), \( -4 \), \( -3 \) and \( -1 \).
7If \( (x + 3) \) is a factor of the polynomial \( 3x^3 + mx^2 - 7x - 3 \), what is the value of m? Choose from 2, 3, 5 and 7.
8State the factor theorem. Using it, prove that \( x - a \) is a factor of f(x) when f(x) is exactly divisible by \( x - a \).
9If \( x - a \) is a factor of a polynomial f(x) of degree n, where \( n \ge 1 \), prove that the remainder \( f(a) = 0 \).
10Using the factor theorem, prove that \( x - 3 \) is a factor of \( f(x) = 2x^2 - 11x + 15 \).
11Using the factor theorem, prove that \( 2x - 1 \) is a factor of \( f(x) = 2x^3 - 7x^2 + 13x - 5 \).
12Using the factor theorem, prove that \( 2x + 1 \) is a factor of \( f(x) = 2x^3 - 5x^2 + 7x + 5 \).
13If \( x + 2 \) is a factor of the polynomial \( f(x) = 4x^2 + kx + 8 \), find the value of k.
14If \( x + 1 \) is a factor of the polynomial \( f(x) = 2x^3 - kx^2 - 8x + 5 \), find the value of k.
15What number must be added to \( f(x) = x^3 + 6x^2 + 11x + 1 \) so that \( x + 1 \) becomes a factor of it?
16What number must be added to \( f(x) = x^3 + x^2 - 17x + 7 \) so that \( x - 3 \) becomes a factor of it?
17What number must be subtracted from \( f(x) = x^3 + 9x^2 + 26x + 30 \) so that \( x + 4 \) becomes a factor of it?
18Is \( x + 2 \) a factor of \( x^3 + 8 \)? Give a reason.
19Show that \( x - 1 \) is not a factor of \( f(x) = x^2 + 1 \), and find the remainder when f(x) is divided by \( x - 1 \).
20If both \( x - 1 \) and \( x + 2 \) are factors of \( f(x) = x^3 + ax^2 + bx - 6 \), find the values of a and b.
21Explain the difference between the factor theorem and its converse.