Solutions to Multiply Polynomials Grade 9

Solutions to questions on how to multiply polynomials are presented along with their detailed explanations.

Solutions to the Questions in multiply polynomials

1. Multiply the following polynomials.
   
   
   1. ) \( ( - 2) (- 2 x) = (-2)(-2)(x) = 4 x \)
      
   2. ) \( (x) (x + 1) = x (x) + x (1) = x^2 + x\) \( \) \( \) \( \) \( \) \( \) \( \) \( \)
      
   3. ) \( - x^2 (- x + 1) \\ = -x^2 (-x) -x^2 (1) \\ = -1(-1)(x^2 x) -1(1)(x^2) = x^3 - x^2\)
      
   4. ) \( (- 4 x^3 - x) (2x - 1) \\ = - 4 x^3(2x-1) - x (2x - 1) \\ = - 4 x^3(2x) - 4 x^3(-1) - x(2x) - x(-1) \\ = -8x^4 + 4x^3 -2x^2 + x \)
      
   5. ) \( (- 4 x^3 - y) (2x - y) \\ = - 4 x^3 (2x - y) - y(2x - y) \\ = - 4 x^3(2x) - 4 x^3(-y) - y(2x) - y(-y) \\ = -8x^4 + 4x^3 y -2x y +y^2 \)
      
   6. ) \( (- 7 x^2 - 2x + 3) (2x^2 - x + 2) \\ = - 7 x^2(2x^2 - x + 2) - 2x(2x^2 - x + 2) + 3(2x^2 - x + 2) \\ = - 7 x^2 (2x^2) - 7 x^2( - x )- 7 x^2(+2) \\ \qquad \qquad\qquad - 2x (2x^2)- 2x (- x )- 2x(+2) + 3(2x^2) + 3(-x) + 3(+2) \\ = - 14 x^4 + 7 x^3 - 14 x^2 - 4x^3 + 2x^2 - 4x - 3x + 6x^2 + 6 \\ = - 14 x^4 + 3 x^3 - 6x^2 - 7x + 6 \)
      
   7. ) \( (- \dfrac{1}{3} x^2 + 4) (- \dfrac{1}{2} x + 9) \\ = - \dfrac{1}{3} x^2 (- \dfrac{1}{2} x + 9) + 4 (- \dfrac{1}{2} x + 9) \\ = - \dfrac{1}{3} x^2(- \dfrac{1}{2} x) - \dfrac{1}{3} x^2(9) + 4(- \dfrac{1}{2} x ) + 4(9) \\ = \dfrac{1}{6} x^3 - 3x^2 - 2x+ 36 \)
   
   
2. Use multiplication and addition of polynomials to write as a single polynomial.
   The usual order of operations is followed: multiplication of polynomials and then addition/subtraction.
   
   
   1. ) \( (2x-1)(3x-2) + 3x - 9 \\ = 2x(3x-2) - 1(3x-2) + 3x - 9 \\ = 6x^2 - 4x -3x + 2 +3x -9 = 6x^2 -4x - 7\)
      
   2. ) \( -(2x + 2) - (2x - 1)(x - 3) \\ = -1(2x) - 1(+2) - (2x (x - 3) - 1 (x - 3)) \\ = -2x - 2 - 1(2x^2-6x-x+3) \\ = -2x - 2 - 2x^2 + 7x - 3 \\ = -2x^2 + 5x - 5 \)
      
   3. ) \( (-3x - 2)(y - 3) + (x - 5)(y - 6) \\ = -3x(y - 3) -2(y - 3) + x (y - 6) - 5 (y - 6) \\ = -3xy + 9x -2y + 6 + xy - 6x -5y + 30 \\ = 3x-2xy-7y + 36 \)
      
   
   
3. Use multiplication of polynomials to expand the following.
   
   
   1. ) \( (x + 3 y)^2 = (x + 3y)(x+3y) \\ = x(x+3y) + 3y(x+3y) \\ = x^2 + 3xy + 3xy + 9y^2 \\ = x^2 + 6xy + 9y^2 \)
      
   2. ) \( (2 x - y)^2 = (2 x - y)(2 x - y) \\ = 2 x(2 x - y) - y(2 x - y) \\ = 4x^2 - 2xy - 2yx + y^2 \\ = 4x^2 - 4xy+ y^2 \)
      
   3. ) \( (x - y)(x +y) = x(x +y) - y(x +y) \\ = x^2 + xy - yx - y^2 \\ = x^2 - y^2 \)
      
   4. ) \( (x - 3)^3 = (x - 3) (x - 3) (x - 3) \\ = (x - 3)( x(x - 3) - 3(x - 3) ) \\ = (x - 3)(x^2 - 3x - 3x + 9) \\ = (x - 3)(x^2 - 6x + 9) \\ = x(x^2 - 6x + 9) - 3(x^2 - 6x + 9) \\ = x^3 - 6x^2 + 9x - 3x^2 + 18x - 27 \\ = x^3 - 9x^2 + 27x - 27 \)

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