Collection of Maths Problems

Operations on polynomials

Task number: 2630

• Variant 1

  \(p(x)=5x^3+3x^2+4x+3\), \(q(x)=3x^2-1x+5\) over \(\mathbb R\)

• Resolution

  The sum and the difference by components:

  \((p+q)(x)=5x^3+6x^2+3x+8\)
  \((p-q)(x)=5x^3+5x-2\)

  The product:

  \((pq)(x)=15x^5+4x^4+34x^3+20x^2+17x+2\)

  Quotient:

  \(p(x)=q(x)\cdot(\frac{5}{3}x+\frac{14}{9})-(\frac{25}{9}x+\frac{43}{9})\).

• Variant 2

  \(p(x)=3x^3+2x^2+4x+1\), \(q(x)=x^2+2x+2\) over \(\mathbb Z_5\)

• Resolution

  \((p+q)(x)=3x^3+3x^2+x+3\)

  \((p-q)(x)=3x^3+x^2+2x+4\)

  \((pq)(x)=3x^5+3x^4+4x^3+3x^2+2\)

  The powers greater than 5 could be eliminated by the Fermat's Little Theorem and get a polynomial with the same values for all \(x\in \mathbb Z_5\):

  \(3x^5+3x^4+4x^3+3x^2+2 \to 3x^4+4x^3+3x^2+3x+2\)

  \(p(x)=q(x)\cdot(3x+1)+(x+4)\).