Question

find the product. write the answer in standard form.
1
$(3x + x^3)(x^3 - 5x)$
2
$(1 + 2x)(6x - 7x^2)$
3
$(8x + 8)^2$
4
$(12x + 14)(7x + 11x)$

Explanation:

Problem 1: \((3x + x^3)(x^3 - 5x)\)

Step 1: Apply the distributive property (FOIL method)

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

Step 2: Simplify each term

• \(3x \cdot x^3 = 3x^{1 + 3} = 3x^4\)
• \(3x \cdot (-5x) = -15x^{1 + 1} = -15x^2\)
• \(x^3 \cdot x^3 = x^{3 + 3} = x^6\)
• \(x^3 \cdot (-5x) = -5x^{3 + 1} = -5x^4\)

Step 3: Combine like terms

Combine the \(x^4\) terms: \(3x^4 - 5x^4 = -2x^4\)
The other terms are \(-15x^2\) and \(x^6\).
So the polynomial in standard form (descending powers of \(x\)) is: \(x^6 - 2x^4 - 15x^2\)

Step 1: Apply the distributive property (FOIL method)

Multiply each term in the first polynomial by each term in the second polynomial:
\(1 \cdot 6x + 1 \cdot (-7x^2) + 2x \cdot 6x + 2x \cdot (-7x^2)\)

Step 2: Simplify each term

• \(1 \cdot 6x = 6x\)
• \(1 \cdot (-7x^2) = -7x^2\)
• \(2x \cdot 6x = 12x^{1 + 1} = 12x^2\)
• \(2x \cdot (-7x^2) = -14x^{1 + 2} = -14x^3\)

Step 3: Combine like terms

Combine the \(x^2\) terms: \(-7x^2 + 12x^2 = 5x^2\)
The other terms are \(6x\) and \(-14x^3\).
So the polynomial in standard form (descending powers of \(x\)) is: \(-14x^3 + 5x^2 + 6x\)

Step 1: Recall the formula for squaring a binomial \((a + b)^2 = a^2 + 2ab + b^2\)

Here, \(a = 8x\) and \(b = 8\).

Step 2: Apply the formula

• \(a^2 = (8x)^2 = 64x^2\)
• \(2ab = 2 \cdot 8x \cdot 8 = 128x\)
• \(b^2 = 8^2 = 64\)

Step 3: Combine the terms

So \((8x + 8)^2 = 64x^2 + 128x + 64\)

Answer:

\(x^6 - 2x^4 - 15x^2\)

Problem 2: \((1 + 2x)(6x - 7x^2)\)