Question

1. $(x^2 + 2x + 6)(5x^2 + 2)$

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$$\begin{array}{r} x^2 + 2x + 6\\\\ \\times\\ \\underline{5x^2\\ \\ \\ \\ + 2}\\\\ \\end{array}$$

$

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

$\

$$\begin{array}{r} 2x^2 + 3x + 4\\\\ \\times\\ \\underline{4x^2\\ \\ \\ \\ - 3}\\\\ \\end{array}$$

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1. practice: summarizing

fill in the blanks to compare and contrast multiplying binomials and trinomials.
9.4.1 study: multiplying polynomials
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Explanation:

To solve the multiplication of the given polynomials, we'll use the distributive property (also known as the FOIL method extended for polynomials with more terms). Let's solve each problem one by one.

Problem 1: \((x^2 + 2x + 6)(5x^2 + 2)\)

Step 1: Distribute each term in the first polynomial to each term in the second polynomial.

• Multiply \(x^2\) by \(5x^2\): \(x^2 \cdot 5x^2 = 5x^{2+2} = 5x^4\)
• Multiply \(x^2\) by \(2\): \(x^2 \cdot 2 = 2x^2\)
• Multiply \(2x\) by \(5x^2\): \(2x \cdot 5x^2 = 10x^{1+2} = 10x^3\)
• Multiply \(2x\) by \(2\): \(2x \cdot 2 = 4x\)
• Multiply \(6\) by \(5x^2\): \(6 \cdot 5x^2 = 30x^2\)
• Multiply \(6\) by \(2\): \(6 \cdot 2 = 12\)

Step 2: Combine like terms.

Now, we have the terms: \(5x^4\), \(2x^2\), \(10x^3\), \(4x\), \(30x^2\), \(12\)

Combine the \(x^2\) terms: \(2x^2 + 30x^2 = 32x^2\)

So, putting it all together: \(5x^4 + 10x^3 + 32x^2 + 4x + 12\)

Problem 2: \((2x^2 + 3x + 4)(4x^2 - 3)\)

Step 1: Distribute each term in the first polynomial to each term in the second polynomial.

• Multiply \(2x^2\) by \(4x^2\): \(2x^2 \cdot 4x^2 = 8x^{2+2} = 8x^4\)
• Multiply \(2x^2\) by \(-3\): \(2x^2 \cdot (-3) = -6x^2\)
• Multiply \(3x\) by \(4x^2\): \(3x \cdot 4x^2 = 12x^{1+2} = 12x^3\)
• Multiply \(3x\) by \(-3\): \(3x \cdot (-3) = -9x\)
• Multiply \(4\) by \(4x^2\): \(4 \cdot 4x^2 = 16x^2\)
• Multiply \(4\) by \(-3\): \(4 \cdot (-3) = -12\)

Step 2: Combine like terms.

Now, we have the terms: \(8x^4\), \(-6x^2\), \(12x^3\), \(-9x\), \(16x^2\), \(-12\)

Combine the \(x^2\) terms: \(-6x^2 + 16x^2 = 10x^2\)

So, putting it all together: \(8x^4 + 12x^3 + 10x^2 - 9x - 12\)

Final Answers:

1. \(\boldsymbol{(x^2 + 2x + 6)(5x^2 + 2) = 5x^4 + 10x^3 + 32x^2 + 4x + 12}\)
2. \(\boldsymbol{(2x^2 + 3x + 4)(4x^2 - 3) = 8x^4 + 12x^3 + 10x^2 - 9x - 12}\)

Answer:

To solve the multiplication of the given polynomials, we'll use the distributive property (also known as the FOIL method extended for polynomials with more terms). Let's solve each problem one by one.

Problem 1: \((x^2 + 2x + 6)(5x^2 + 2)\)

Step 1: Distribute each term in the first polynomial to each term in the second polynomial.

• Multiply \(x^2\) by \(5x^2\): \(x^2 \cdot 5x^2 = 5x^{2+2} = 5x^4\)
• Multiply \(x^2\) by \(2\): \(x^2 \cdot 2 = 2x^2\)
• Multiply \(2x\) by \(5x^2\): \(2x \cdot 5x^2 = 10x^{1+2} = 10x^3\)
• Multiply \(2x\) by \(2\): \(2x \cdot 2 = 4x\)
• Multiply \(6\) by \(5x^2\): \(6 \cdot 5x^2 = 30x^2\)
• Multiply \(6\) by \(2\): \(6 \cdot 2 = 12\)

Step 2: Combine like terms.

Now, we have the terms: \(5x^4\), \(2x^2\), \(10x^3\), \(4x\), \(30x^2\), \(12\)

Combine the \(x^2\) terms: \(2x^2 + 30x^2 = 32x^2\)

So, putting it all together: \(5x^4 + 10x^3 + 32x^2 + 4x + 12\)

Problem 2: \((2x^2 + 3x + 4)(4x^2 - 3)\)

Step 1: Distribute each term in the first polynomial to each term in the second polynomial.

• Multiply \(2x^2\) by \(4x^2\): \(2x^2 \cdot 4x^2 = 8x^{2+2} = 8x^4\)
• Multiply \(2x^2\) by \(-3\): \(2x^2 \cdot (-3) = -6x^2\)
• Multiply \(3x\) by \(4x^2\): \(3x \cdot 4x^2 = 12x^{1+2} = 12x^3\)
• Multiply \(3x\) by \(-3\): \(3x \cdot (-3) = -9x\)
• Multiply \(4\) by \(4x^2\): \(4 \cdot 4x^2 = 16x^2\)
• Multiply \(4\) by \(-3\): \(4 \cdot (-3) = -12\)

Step 2: Combine like terms.

Now, we have the terms: \(8x^4\), \(-6x^2\), \(12x^3\), \(-9x\), \(16x^2\), \(-12\)

Combine the \(x^2\) terms: \(-6x^2 + 16x^2 = 10x^2\)

So, putting it all together: \(8x^4 + 12x^3 + 10x^2 - 9x - 12\)

Final Answers:

1. \(\boldsymbol{(x^2 + 2x + 6)(5x^2 + 2) = 5x^4 + 10x^3 + 32x^2 + 4x + 12}\)
2. \(\boldsymbol{(2x^2 + 3x + 4)(4x^2 - 3) = 8x^4 + 12x^3 + 10x^2 - 9x - 12}\)