Question

find the difference of the polynomials given below and classify it in terms of degree and number of terms
$3n^2(n^2 + 4n - 5) - (2n^2 - n^4 + 3)$
a. $3^{rd}$ degree polynomial with 4 terms
b. $3^{rd}$ degree polynomial with 5 terms
c. $4^{th}$ degree polynomial with 4 terms
d. $4^{th}$ degree polynomial with 5 terms

Explanation:

Step1: Expand the first polynomial

Using the distributive property (multiply \(3n^2\) with each term inside the parentheses):
\(3n^2(n^2 + 4n - 5) = 3n^2 \cdot n^2 + 3n^2 \cdot 4n - 3n^2 \cdot 5 = 3n^4 + 12n^3 - 15n^2\)

Step2: Distribute the negative sign

Apply the negative sign to each term in \((2n^2 - n^4 + 3)\):
\(-(2n^2 - n^4 + 3) = -2n^2 + n^4 - 3\)

Step3: Combine the expanded polynomials

Add the results from Step 1 and Step 2:
\((3n^4 + 12n^3 - 15n^2) + (n^4 - 2n^2 - 3)\)

Step4: Combine like terms

• For \(n^4\): \(3n^4 + n^4 = 4n^4\)
• For \(n^3\): \(12n^3\) (no like terms to combine)
• For \(n^2\): \(-15n^2 - 2n^2 = -17n^2\)
• Constant term: \(-3\) (no like terms to combine)

The simplified polynomial is \(4n^4 + 12n^3 - 17n^2 - 3\).

Step5: Classify the polynomial

• Degree: The highest power of \(n\) is \(4\) (from \(4n^4\)), so it is a \(4^{\text{th}}\) degree polynomial.
• Number of terms: The terms are \(4n^4\), \(12n^3\), \(-17n^2\), and \(-3\) (4 terms total).

Answer:

C. \(4^{\text{th}}\) degree polynomial with 4 terms