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.

Question

Rewrite this in two paragraphs Two examples of polynomial expressions are \(A(x) = 4x^5 - 3x^3 + 2x - 7\) (which is of degree 5) and \(B(x) = 2x^3 + 6x^2 + 1\) (which is of degree 3). When the two polynomials are added together to find their sum, you have to combine the like terms. The resulting polynomial is \[C(x) = A(x) + B(x) = 4x^5 + ( -3x^3 + 2x^3) + 6x^2 + (2x) - 7 + 1 = 4x^5 - x^3 + 6x^2 + 2x - 6.\], and the highest degree term in this sum is \(4x^5\), which gives the polynomial \(C(x)\) a degree of 5.

To find the first and last terms of a polynomial, I looked for the leading term and the constant term.

Answer 1

Two examples of polynomial expressions are \(A(x) = 4x^4 - 3x^3 + 2x - 7\) (which is of degree 4) and \(B(x) = 2x^3 + 6x^2 + 1\) (which is of degree 3). To find the sum of these two polynomials, we combine like terms. The resulting polynomial is \[C(x) = A(x) + B(x) = 4x^4 + (-3x^3 + 2x^3) + 6x^2 + (2x) + (-7 + 1) = 4x^4 - x^3 + 6x^2 + 2x - 6.\] The highest degree term in this sum is \(4x^4\), which gives the polynomial \(C(x)\) a degree of 4.

Next, we can compute the difference, product, and quotient of these two polynomials. The difference is \(D(x) = A(x) - B(x) = 4x^4 + ( -3x^3 - 2x^3) + 6x^2 + (2x) + (-7 - 1) = 4x^4 - 5x^3 + 6x^2 + 2x - 8.\) The product is \(E(x) = A(x) \cdot B(x)\), which gives us a polynomial with a degree of 7. For the quotient, since \(A(x)\) has a higher degree, we perform polynomial long division: \(Q(x) = \frac{A(x)}{B(x)}\). The result will be a polynomial with a degree of 1, as the leading term of \(A(x)\) divided by the leading term of \(B(x)\) results in an \(x\) term. The exact expressions for \(E(x)\) and \(Q(x)\) can be derived more precisely through multiplication and division, respectively.