Enrichment, Extension, and Application

Question 1

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Exercises

  1. Suppose that \( m \) and \( n \) are real numbers for which the three (not necessarily distinct) roots of \( x^3 - mx^2 + nx - 1 = 0 \) are \( m\), \(n \), and \( 1 \). What is the value of \( m + n + 1 \)?
    1. Show that the polynomial \( f(x) = ax^3 + bx^2 + cx + d \), where \( a + b + c + d = 0 \) is divisible by \( x - 1 \).
    2. Solve \( 2x^3 - 3x^2 + 1 \le 0 \).
    3. Find the sum of the coefficients in the expansion of \( g(x) = (x + 2)(x^2 + 2x + 1)^2. \)
  2. Determine all values of \( t \) such that the roots of \( t(x - 1)(x - 2) = x \) are real.
  3. Consider all equations \( x^2 + bx + c = 0 \) where \( b \) and \( c \) are integers between \( 1 \) and \( 8 \) inclusive. Determine how many of these equations have no real roots.
  4. Find all values of \( x \) that solve the equation \[ (x^2 - 3x + 1)^2 - 3(x^2 - 3x + 1) + 1 = x\]
  5. If \( ax^3 + bx + c \), with \( a \ne 0, c \ne 0 \), has a factor of the form \( x^2 + px + 1 \), show that \( a^2 - c^2 = ab \).
  6. In the equation \( x^4 + bx^3 + cx^2 - 630 = 0\), \(a\), \(b \), and \( c \) are integers. If \( p \) is a prime number which is a root of the equation, then determine the value of \( p \).
  7. If \( k \) is a positive integer, determine the smallest positive value of \( k \) for which the equation \[x^2 + kx = 4y^2 - 4y + 1\] has a solution \( (x,y) \), where \( x \) and \( y \) are both integers.
  8. The remainder when \( f(x) = x^5 - 2x^4 + ax^3 - x^2 + bx - 2 \) is divided by \( x + 1 \) is \( - 7 \). When \( f(x) \) is divided by \( x - 2 \), the remainder is \( 32 \). Determine the remainder when \( f(x) \) is divided by \( x - 1 \).
  9. Find all pairs of primes \( q \) and \( r \) so that the equation \( 5x^2 - qx + r = 0 \) has distinct rational roots.
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