Question

find the product.
$(2x^5 + 3x^3)(x^5 + x^3)$
$?x^{10} + \quadx^8 + \quadx^6$

Explanation:

Step1: Apply distributive property (FOIL)

Multiply \(2x^5\) with each term in \((x^5 + x^3)\) and \(3x^3\) with each term in \((x^5 + x^3)\):
\(2x^5 \cdot x^5 + 2x^5 \cdot x^3 + 3x^3 \cdot x^5 + 3x^3 \cdot x^3\)

Step2: Use exponent rule \(a^m \cdot a^n = a^{m + n}\)

For \(2x^5 \cdot x^5\): \(2x^{5 + 5}=2x^{10}\)
For \(2x^5 \cdot x^3\): \(2x^{5 + 3}=2x^8\)
For \(3x^3 \cdot x^5\): \(3x^{3 + 5}=3x^8\)
For \(3x^3 \cdot x^3\): \(3x^{3 + 3}=3x^6\)

Step3: Combine like terms for \(x^8\)

\(2x^8 + 3x^8 = 5x^8\)

So the expanded form is \(2x^{10}+5x^8 + 3x^6\)

Answer:

For \(x^{10}\) term: \(2\); for \(x^8\) term: \(5\); for \(x^6\) term: \(3\) (In the order of the boxes: first box \(2\), second box \(5\), third box \(3\))