Question

1. simplify: $(2x - 5)^2$
2. simplify: $(x + 2)(x^2 + 3x + 1)$
3. simplify: $(x - 3)(x^2 - 2x + 5)$
4. simplify: $(2x + 1)(3x^2 - 4x - 2)$
5. simplify: $(x^2 + 2)(x^2 - 3)$

Explanation:

Problem 12: Simplify \((2x - 5)^2\)

Step 1: Apply the square of a binomial formula \((a - b)^2 = a^2 - 2ab + b^2\)

Here, \(a = 2x\) and \(b = 5\). So we have:
\((2x)^2 - 2\times(2x)\times5 + 5^2\)

Step 2: Simplify each term

• \((2x)^2 = 4x^2\)
• \(2\times(2x)\times5 = 20x\)
• \(5^2 = 25\)

Putting it all together: \(4x^2 - 20x + 25\)

Step 1: Distribute \(x\) and \(2\) into the trinomial

\(x(x^2 + 3x + 1) + 2(x^2 + 3x + 1)\)

Step 2: Multiply each term

• \(x(x^2 + 3x + 1) = x^3 + 3x^2 + x\)
• \(2(x^2 + 3x + 1) = 2x^2 + 6x + 2\)

Step 3: Combine like terms

\(x^3 + (3x^2 + 2x^2) + (x + 6x) + 2 = x^3 + 5x^2 + 7x + 2\)

Step 1: Distribute \(x\) and \(-3\) into the trinomial

\(x(x^2 - 2x + 5) - 3(x^2 - 2x + 5)\)

Step 2: Multiply each term

• \(x(x^2 - 2x + 5) = x^3 - 2x^2 + 5x\)
• \(-3(x^2 - 2x + 5) = -3x^2 + 6x - 15\)

Step 3: Combine like terms

\(x^3 + (-2x^2 - 3x^2) + (5x + 6x) - 15 = x^3 - 5x^2 + 11x - 15\)

Answer:

\(4x^2 - 20x + 25\)

Problem 13: Simplify \((x + 2)(x^2 + 3x + 1)\)