Question

find the product. then list the degree. circle b if it is a binomial, t if it is a trinomial or p if it is a polynomial with more than 3 terms.
1
$(x^2 + 2x)(2x - x^3)$
degree = ______ b t p
2
$(2x^2 + 11)(2x^2 - 11)$
degree = ______ b t p
3
$(3x^2 - 5)(4 + 4x^2)$
degree = ______ b t p

Explanation:

Problem 1: \((x^2 + 2x)(2x - x^3)\)

Step 1: Apply distributive property (FOIL)

Multiply each term in the first binomial by each term in the second binomial:
\(x^2 \cdot 2x + x^2 \cdot (-x^3) + 2x \cdot 2x + 2x \cdot (-x^3)\)

Step 2: Simplify each term

Calculate the product of each pair:
\(2x^3 - x^5 + 4x^2 - 2x^4\)

Step 3: Rearrange terms by degree

Arrange in descending order of exponents:
\(-x^5 - 2x^4 + 2x^3 + 4x^2\)

Step 4: Determine the degree

The degree of a polynomial is the highest power of \(x\), which is \(5\) here.

Step 5: Classify the polynomial

The polynomial has \(4\) terms, so it's a polynomial with more than \(3\) terms (P).

Step 1: Apply difference of squares formula

The formula \((a + b)(a - b) = a^2 - b^2\) where \(a = 2x^2\) and \(b = 11\):
\((2x^2)^2 - 11^2\)

Step 2: Simplify each term

Calculate the squares:
\(4x^4 - 121\)

Step 3: Determine the degree

The highest power of \(x\) is \(4\), so the degree is \(4\).

Step 4: Classify the polynomial

The polynomial has \(2\) terms, so it's a binomial (B).

Step 1: Apply distributive property (FOIL)

Multiply each term in the first binomial by each term in the second binomial:
\(3x^2 \cdot 4 + 3x^2 \cdot 4x^2 - 5 \cdot 4 - 5 \cdot 4x^2\)

Step 2: Simplify each term

Calculate the product of each pair:
\(12x^2 + 12x^4 - 20 - 20x^2\)

Step 3: Combine like terms

Combine the \(x^2\) terms:
\(12x^4 - 8x^2 - 20\)

Step 4: Determine the degree

The highest power of \(x\) is \(4\), so the degree is \(4\).

Step 5: Classify the polynomial

The polynomial has \(3\) terms, so it's a trinomial (T).

Answer:

Product: \(-x^5 - 2x^4 + 2x^3 + 4x^2\)
degree = \(5\)
Circle: P

Problem 2: \((2x^2 + 11)(2x^2 - 11)\)