Use distributive property to find the product of

Question

(2𝑥
2
+ 3𝑥 + 2)(5𝑥
2 − 10𝑥 + 3)

Answer 1

To find the product of the polynomials \((2x^2 + 3x + 2)(5x^2 - 10x + 3)\) using the distributive property, we will multiply each term in the first polynomial by each term in the second polynomial.

Let's denote the first polynomial as \(A = 2x^2 + 3x + 2\) and the second polynomial as \(B = 5x^2 - 10x + 3\).

Step 1: Multiply each term in \(A\) by each term in \(B\)

1. Multiplying \(2x^2\) by each term in \(B\):

   • \(2x^2 \cdot 5x^2 = 10x^4\)
   • \(2x^2 \cdot (-10x) = -20x^3\)
   • \(2x^2 \cdot 3 = 6x^2\)

   So, the result of multiplying \(2x^2\) by \(B\) is: \[ 10x^4 - 20x^3 + 6x^2 \]

2. Multiplying \(3x\) by each term in \(B\):

   • \(3x \cdot 5x^2 = 15x^3\)
   • \(3x \cdot (-10x) = -30x^2\)
   • \(3x \cdot 3 = 9x\)

   So, the result of multiplying \(3x\) by \(B\) is: \[ 15x^3 - 30x^2 + 9x \]

3. Multiplying \(2\) by each term in \(B\):

   • \(2 \cdot 5x^2 = 10x^2\)
   • \(2 \cdot (-10x) = -20x\)
   • \(2 \cdot 3 = 6\)

   So, the result of multiplying \(2\) by \(B\) is: \[ 10x^2 - 20x + 6 \]

Step 2: Combine all results

Now add all the results together:

\[ (10x^4 - 20x^3 + 6x^2) + (15x^3 - 30x^2 + 9x) + (10x^2 - 20x + 6) \]

Step 3: Group like terms

Now, let's group the like terms:

• For \(x^4\): \(10x^4\)
• For \(x^3\): \(-20x^3 + 15x^3 = -5x^3\)
• For \(x^2\): \(6x^2 - 30x^2 + 10x^2 = -14x^2\)
• For \(x\): \(9x - 20x = -11x\)
• Constant: \(6\)

Step 4: Write the final result

Putting it all together, we have: \[ 10x^4 - 5x^3 - 14x^2 - 11x + 6 \]

So, the product of \((2x^2 + 3x + 2)(5x^2 - 10x + 3)\) is: \[ \boxed{10x^4 - 5x^3 - 14x^2 - 11x + 6} \]