Question

ats
multiply polynomials bi by tri
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find each product.

1. (8a - a)(7a² + 4a - 1)
2. (-5x - 2)(3x² + 2x + 5)
3. (-5x - 3)(-2x² + 3x + 1)
4. (-4n - 4)(8n² + 8n + 1)

Explanation:

Problem 1: \((8a - 7)(7a^2 + 4a - 1)\)

Step 1: Distribute \(8a\)

Multiply \(8a\) with each term in \(7a^2 + 4a - 1\):
\(8a \times 7a^2 = 56a^3\), \(8a \times 4a = 32a^2\), \(8a \times (-1) = -8a\)
So, \(8a(7a^2 + 4a - 1) = 56a^3 + 32a^2 - 8a\)

Step 2: Distribute \(-7\)

Multiply \(-7\) with each term in \(7a^2 + 4a - 1\):
\(-7 \times 7a^2 = -49a^2\), \(-7 \times 4a = -28a\), \(-7 \times (-1) = 7\)
So, \(-7(7a^2 + 4a - 1) = -49a^2 - 28a + 7\)

Step 3: Combine like terms

Add the results from Step 1 and Step 2:
\(56a^3 + 32a^2 - 8a - 49a^2 - 28a + 7\)
Combine \(32a^2 - 49a^2 = -17a^2\) and \(-8a - 28a = -36a\)
Final: \(56a^3 - 17a^2 - 36a + 7\)

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

Step 1: Distribute \(-5x\)

Multiply \(-5x\) with each term in \(3x^2 + 2x + 5\):
\(-5x \times 3x^2 = -15x^3\), \(-5x \times 2x = -10x^2\), \(-5x \times 5 = -25x\)
So, \(-5x(3x^2 + 2x + 5) = -15x^3 - 10x^2 - 25x\)

Step 2: Distribute \(-2\)

Multiply \(-2\) with each term in \(3x^2 + 2x + 5\):
\(-2 \times 3x^2 = -6x^2\), \(-2 \times 2x = -4x\), \(-2 \times 5 = -10\)
So, \(-2(3x^2 + 2x + 5) = -6x^2 - 4x - 10\)

Step 3: Combine like terms

Add the results from Step 1 and Step 2:
\(-15x^3 - 10x^2 - 25x - 6x^2 - 4x - 10\)
Combine \(-10x^2 - 6x^2 = -16x^2\) and \(-25x - 4x = -29x\)
Final: \(-15x^3 - 16x^2 - 29x - 10\)

Problem 3: \((-5x - 3)(-2x^2 + 3x + 1)\)

Step 1: Distribute \(-5x\)

Multiply \(-5x\) with each term in \(-2x^2 + 3x + 1\):
\(-5x \times (-2x^2) = 10x^3\), \(-5x \times 3x = -15x^2\), \(-5x \times 1 = -5x\)
So, \(-5x(-2x^2 + 3x + 1) = 10x^3 - 15x^2 - 5x\)

Step 2: Distribute \(-3\)

Multiply \(-3\) with each term in \(-2x^2 + 3x + 1\):
\(-3 \times (-2x^2) = 6x^2\), \(-3 \times 3x = -9x\), \(-3 \times 1 = -3\)
So, \(-3(-2x^2 + 3x + 1) = 6x^2 - 9x - 3\)

Step 3: Combine like terms

Add the results from Step 1 and Step 2:
\(10x^3 - 15x^2 - 5x + 6x^2 - 9x - 3\)
Combine \(-15x^2 + 6x^2 = -9x^2\) and \(-5x - 9x = -14x\)
Final: \(10x^3 - 9x^2 - 14x - 3\)

Problem 4: \((-4n - 4)(8n^2 + 8n + 1)\)

Answer:

s:

1. \(\boldsymbol{56a^3 - 17a^2 - 36a + 7}\)
2. \(\boldsymbol{-15x^3 - 16x^2 - 29x - 10}\)
3. \(\boldsymbol{10x^3 - 9x^2 - 14x - 3}\)
4. \(\boldsymbol{-32n^3 - 64n^2 - 36n - 4}\)