Question

math 2 chapter 2.2 multiplying polynomials
use the f.o.i.l. method to find each product.

1. ((x + 1)(x - 7)) 2) ((x + 3)(x + 5)) 3) ((x - 2)(x + 10))
2. ((x + 8)(x + 2)) 5) ((3x + 2)(x - 1)) 6) ((x - 4)(2x + 1))
3. ((x^2 + 8x + 2)(x + 3)) 8) ((2x^2 + 3)(4x - 1)) 9) ((3x^2 + 2x + 1)(5x - 2))

find each product:

1. ((x + 4)^2) 11) ((x - 4)^2) 12) ((x + 16)^2)
2. ((x + 14)^2) 14) ((3x + 4)^2) 15) ((5x + 2)^2)
3. ((4x + 1)^2) 17) ((x - 6)(x + 6)) 18) ((2x - 5)(2x + 5))

Explanation:

Step1: Apply F.O.I.L. to \((x + 1)(x - 7)\)

Multiply First terms: \(x \times x = x^2\)
Multiply Outer terms: \(x \times (-7) = -7x\)
Multiply Inner terms: \(1 \times x = x\)
Multiply Last terms: \(1 \times (-7) = -7\)

Step2: Combine like terms

\(x^2 - 7x + x - 7 = x^2 - 6x - 7\)

Step1: Apply F.O.I.L. to \((x + 3)(x + 5)\)

First: \(x \times x = x^2\)
Outer: \(x \times 5 = 5x\)
Inner: \(3 \times x = 3x\)
Last: \(3 \times 5 = 15\)

Step2: Combine like terms

\(x^2 + 5x + 3x + 15 = x^2 + 8x + 15\)

Step1: Apply F.O.I.L. to \((x - 2)(x + 10)\)

First: \(x \times x = x^2\)
Outer: \(x \times 10 = 10x\)
Inner: \(-2 \times x = -2x\)
Last: \(-2 \times 10 = -20\)

Step2: Combine like terms

\(x^2 + 10x - 2x - 20 = x^2 + 8x - 20\)

Step1: Apply F.O.I.L. to \((x + 8)(x + 2)\)

First: \(x \times x = x^2\)
Outer: \(x \times 2 = 2x\)
Inner: \(8 \times x = 8x\)
Last: \(8 \times 2 = 16\)

Step2: Combine like terms

\(x^2 + 2x + 8x + 16 = x^2 + 10x + 16\)

Step1: Apply F.O.I.L. to \((3x + 2)(x - 1)\)

First: \(3x \times x = 3x^2\)
Outer: \(3x \times (-1) = -3x\)
Inner: \(2 \times x = 2x\)
Last: \(2 \times (-1) = -2\)

Step2: Combine like terms

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

Step1: Apply F.O.I.L. to \((x - 4)(2x + 1)\)

First: \(x \times 2x = 2x^2\)
Outer: \(x \times 1 = x\)
Inner: \(-4 \times 2x = -8x\)
Last: \(-4 \times 1 = -4\)

Step2: Combine like terms

\(2x^2 + x - 8x - 4 = 2x^2 - 7x - 4\)

Step1: Distribute each term in \((x^2 + 3x + 2)\) to \((x + 3)\)

\(x^2(x) + x^2(3) + 3x(x) + 3x(3) + 2(x) + 2(3)\)
\(= x^3 + 3x^2 + 3x^2 + 9x + 2x + 6\)

Step2: Combine like terms

\(x^3 + (3x^2 + 3x^2) + (9x + 2x) + 6 = x^3 + 6x^2 + 11x + 6\)

Step1: Distribute each term in \((2x^2 + 3)\) to \((4x - 1)\)

\(2x^2(4x) + 2x^2(-1) + 3(4x) + 3(-1)\)
\(= 8x^3 - 2x^2 + 12x - 3\)

Step1: Distribute each term in \((3x^2 + 2x + 1)\) to \((5x - 2)\)

\(3x^2(5x) + 3x^2(-2) + 2x(5x) + 2x(-2) + 1(5x) + 1(-2)\)
\(= 15x^3 - 6x^2 + 10x^2 - 4x + 5x - 2\)

Step2: Combine like terms

\(15x^3 + (-6x^2 + 10x^2) + (-4x + 5x) - 2 = 15x^3 + 4x^2 + x - 2\)

Step1: Use \((a + b)^2 = a^2 + 2ab + b^2\) for \((7x + 3)^2\)

\(a = 7x\), \(b = 3\)
\((7x)^2 + 2(7x)(3) + 3^2 = 49x^2 + 42x + 9\)

Step1: Use \((a - b)^2 = a^2 - 2ab + b^2\) for \((x - 4)^2\)

\(a = x\), \(b = 4\)
\(x^2 - 2(x)(4) + 4^2 = x^2 - 8x + 16\)

Step1: Use \((a + b)^2 = a^2 + 2ab + b^2\) for \((x + 16)^2\)

\(a = x\), \(b = 16\)
\(x^2 + 2(x)(16) + 16^2 = x^2 + 32x + 256\)

Step1: Use \((a + b)^2 = a^2 + 2ab + b^2\) for \((3x + 14)^2\)

\(a = 3x\), \(b = 14\)
\((3x)^2 + 2(3x)(14) + 14^2 = 9x^2 + 84x + 196\)

Step1: Use \((a + b)^2 = a^2 + 2ab + b^2\) for \((9x + 4)^2\)

\(a = 9x\), \(b = 4\)
\((9x)^2 + 2(9x)(4) + 4^2 = 81x^2 + 72x + 16\)

Step1: Use \((a + b)^2 = a^2 + 2ab + b^2\) for \((5x + 2)^2\)

\(a = 5x\), \(b = 2\)
\((5x)^2 + 2(5x)(2) + 2^2 = 25x^2 + 20x + 4\)

Step1: Recognize \((a - b)(a + b) = a^2 - b^2\) for \((8 - b)(8 + b)\)

\(a = 8\), \(b = b\)
\(8^2 - b^2 = 64 - b^2\)

Step1: Recognize \((a - b)(a + b) = a^2 - b^2\) for \((2x - 5)(2x + 5)\)

\(a = 2x\), \(b = 5\)
\((2x)^2 - 5^2 = 4x^2 - 25\)

Answer:

s (for selected problems, e.g., 1–6, 10–12, 17–18):

1. \(x^2 - 6x - 7\)
2. \(x^2 + 8x + 15\)
3. \(x^2 + 8x - 20\)
4. \(x^2 + 10x + 16\)
5. \(3x^2 - x - 2\)
6. \(2x^2 - 7x - 4\)
7. \(49x^2 + 42x + 9\)
8. \(x^2 - 8x + 16\)
9. \(x^2 + 32x + 256\)
10. \(64 - b^2\)
11. \(4x^2 - 25\)

(Note: For all problems, follow the same expansion methods—F.O.I.L. for binomials, distribution for polynomials, and special products for squares/differences of squares.)