Polynomials

• What operations can you perform on polynomials algebraically?

  Expand brackets, collect like terms, factorise
• Why can you only collect like terms in polynomials?

  Because they must have the same power
• If \( f(x) = 5x^3 + 2x^2 - 3x + 7 \) and \( g(x) = 3x^3 - 4x^2 + 8x - 5 \), what is \( f(x) + g(x) \)?

  8x^3 - 2x^2 + 5x + 2
• What is the procedure for subtracting polynomials?

  Subtract each term and change signs
• What must you do when multiplying polynomials?

  Multiply every term in the first bracket
• What is the result of multiplying \( (3x^2 - 4x + 7) \) by \( (x^2 + 3x - 5) \)?

  3x^4 + 5x^3 - 20x^2 + 41x - 35
• How do you handle multiple brackets when multiplying polynomials?

  Multiply the first two, then the result by the next
• What are the two methods of algebraic division mentioned?

  Equating coefficients and long division
• How do you divide \( 3x^3 + 4x^2 - 13x + 6 \) by \( x + 3 \) using long division?

  3x^2 - 5x + 2
• What does the factor theorem state?

  If \( f(a) = 0 \), then \( (x - a) \) is a factor
• How do you find a root of the polynomial \( x^3 + x^2 - 11x + 10 = 0 \)?

  Try different numbers until \( f(x) = 0 \)
• What is the maximum number of times a polynomial of order \( n \) can meet the x-axis?

  At most \( n \) times
• How do you find the y-intercept of a polynomial graph?

  Substitute \( x = 0 \) and solve
• How do you find the x-intercept of a polynomial graph?

  Substitute \( y = 0 \) and solve
• How can you find the turning point of a quadratic graph?

  Complete the square
• What are rational expressions?

  Fractions where numerator and denominator are polynomials
• What is the rule for cancelling in rational expressions?

  You can only cancel factors, not terms