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1. $(3x^2 + 4x + 8)(x^2 + 3x)$ chart with rows $x^2$, $3x$ and columns …

Question

  1. $(3x^2 + 4x + 8)(x^2 + 3x)$

chart with rows $x^2$, $3x$ and columns $3x^2$, $4x$, $8$ for filling products
product: _____________

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

chart with rows $x^2$, $-1$ and columns $2x^2$, $x$, $-3$ for filling products
product: _____________

  1. practice: organizing information

fill in the blanks to complete the steps.

how to use the distributive property to multiply polynomials

  • step 1: distribute the first ____________ over the second one.
  • step 2: use the ____________ property again on the smaller products.
  • step 3: collect ________ terms and simplify the answer.

Explanation:

Response
Problem 1

Step1: Multiply \(x^2\) with each term in \(3x^2 + 4x + 8\)

  • \(x^2\times3x^2 = 3x^{2 + 2}=3x^4\)
  • \(x^2\times4x = 4x^{2 + 1}=4x^3\)
  • \(x^2\times8 = 8x^2\)

Step2: Multiply \(3x\) with each term in \(3x^2 + 4x + 8\)

  • \(3x\times3x^2 = 9x^{1 + 2}=9x^3\)
  • \(3x\times4x = 12x^{1 + 1}=12x^2\)
  • \(3x\times8 = 24x\)

Step3: Add all the products together

\(3x^4 + 4x^3 + 8x^2 + 9x^3 + 12x^2 + 24x\)

Step4: Combine like terms

  • For \(x^3\) terms: \(4x^3+9x^3 = 13x^3\)
  • For \(x^2\) terms: \(8x^2 + 12x^2=20x^2\)

So the product is \(3x^4+13x^3 + 20x^2+24x\)

Step1: Multiply \(x^2\) with each term in \(2x^2 + x - 3\)

  • \(x^2\times2x^2 = 2x^{2+2}=2x^4\)
  • \(x^2\times x = x^{2 + 1}=x^3\)
  • \(x^2\times(-3)=-3x^2\)

Step2: Multiply \(-1\) with each term in \(2x^2 + x - 3\)

  • \(-1\times2x^2=-2x^2\)
  • \(-1\times x=-x\)
  • \(-1\times(-3) = 3\)

Step3: Add all the products together

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

Step4: Combine like terms

  • For \(x^2\) terms: \(-3x^2-2x^2=-5x^2\)

So the product is \(2x^4 + x^3-5x^2 - x + 3\)

Brief Explanations
  • Step 1: When multiplying polynomials using the distributive property, we first distribute the first polynomial over the second one.
  • Step 2: After the first distribution, we use the distributive property again on the smaller products (monomial times polynomial) to expand further.
  • Step 3: After expanding, we collect like terms (terms with the same variable and exponent) and simplify the expression.

Answer:

\(3x^4 + 13x^3 + 20x^2 + 24x\)

Problem 2