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Write two examples of a polynomial expression, each with a minimum of three terms and a degree no larger than four. Give each polynomial expression a different degree. Then find the sum, difference, product, and quotient of your two polynomials. For the quotient, use the polynomial expression with the larger degree as the numerator.

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Let's start with two polynomial expressions:

  1. Polynomial A (degree 4):
    \( P(x) = 3x^4 + 5x^2 - 2 \)

  2. Polynomial B (degree 3):
    \( Q(x) = 2x^3 - x + 7 \)

Now, let's perform the required operations: sum, difference, product, and quotient.

1. Sum of the Polynomials (P + Q)

\[ P(x) + Q(x) = (3x^4 + 5x^2 - 2) + (2x^3 - x + 7) \] \[ = 3x^4 + 2x^3 + 5x^2 - x + 5 \]

2. Difference of the Polynomials (P - Q)

\[ P(x) - Q(x) = (3x^4 + 5x^2 - 2) - (2x^3 - x + 7) \] \[ = 3x^4 - 2x^3 + 5x^2 + x - 9 \]

3. Product of the Polynomials (P * Q)

\[ P(x) \cdot Q(x) = (3x^4 + 5x^2 - 2)(2x^3 - x + 7) \]

To find the product, we distribute each term in \( P(x) \) with every term in \( Q(x) \):

\[ = 3x^4(2x^3) + 3x^4(-x) + 3x^4(7) + 5x^2(2x^3) + 5x^2(-x) + 5x^2(7) - 2(2x^3) - 2(-x) - 2(7) \]

Calculating each term:

  • \( 3x^4 \cdot 2x^3 = 6x^7 \)
  • \( 3x^4 \cdot -x = -3x^5 \)
  • \( 3x^4 \cdot 7 = 21x^4 \)
  • \( 5x^2 \cdot 2x^3 = 10x^5 \)
  • \( 5x^2 \cdot -x = -5x^3 \)
  • \( 5x^2 \cdot 7 = 35x^2 \)
  • \( -2 \cdot 2x^3 = -4x^3 \)
  • \( -2 \cdot -x = 2x \)
  • \( -2 \cdot 7 = -14 \)

Now, combine like terms:

\[ = 6x^7 + (-3x^5 + 10x^5) + (21x^4) + (-5x^3 - 4x^3) + 35x^2 + 2x - 14 \] \[ = 6x^7 + 7x^5 + 21x^4 - 9x^3 + 35x^2 + 2x - 14 \]

4. Quotient of the Polynomials (P / Q)

For the quotient, we divide \( P(x) \) by \( Q(x) \):

The polynomial long division process shows:

  1. Divide the leading terms:
    \( \frac{3x^4}{2x^3} = \frac{3}{2}x \)

  2. Multiply \( Q(x) \) by \( \frac{3}{2}x \) and subtract from \( P(x) \):

    \( \frac{3}{2}x(2x^3 - x + 7) = 3x^4 - \frac{3}{2}x^2 + \frac{21}{2}x \)

    Substituting back: \[ P(x) - (3x^4 - \frac{3}{2}x^2 + \frac{21}{2}x) = 5x^2 - 2 - (-\frac{3}{2}x^2) - \frac{21}{2}x \]
    Now add: \[ = (5 + \frac{3}{2})x^2 - \left(-\frac{21}{2}x + 2 \right) = \left(\frac{10}{2} + \frac{3}{2}\right)x^2 - \frac{21}{2}x - 2 = \frac{13}{2}x^2 - \frac{21}{2}x - 2 \]

So the quotient will be incomplete polynomial.

Final Results

  • Sum: \( 3x^4 + 2x^3 + 5x^2 - x + 5 \)
  • Difference: \( 3x^4 - 2x^3 + 5x^2 + x - 9 \)
  • Product: \( 6x^7 + 7x^5 + 21x^4 - 9x^3 + 35x^2 + 2x - 14 \)
  • Quotient: \( \frac{3}{2}x + \frac{13}{2}x^2 - \frac{21}{2} - 2 \) (expressed as a division of polynomials).

That's how we perform operations on polynomials!