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How do you multiply polynomials?

How do you multiply polynomials?

{
 "voice_prompt": "",
 "manuscript": {
   "title": {
     "text": "How do you multiply polynomials?",
     "audio": "How do you multiply polynomials?"
   },
   "description": {
     "text": "To multiply polynomials, use the distributive property — or the FOIL method for binomials. Let's walk through some examples and understand how each term interacts.",
     "audio": "To multiply polynomials, use the distributive property — or the FOIL method for binomials. Let's walk through some examples and understand how each term interacts."
   },
   "scenes": [
     {
       "text": "Let’s start with a quick reminder: a polynomial is an expression with one or more terms. Each term has a variable raised to a whole number exponent and a coefficient in front.",
       "latex": "a_nx^n + a_{n-1}x^{n-1} + \\dots + a_1x + a_0"
     },
     {
       "text": "Now let's multiply a monomial and a binomial. Consider \\( 2x \\) and \\( 3x + 4 \\).",
       "latex": "2x(3x + 4)"
     },
     {
       "text": "Use the distributive property: multiply \\( 2x \\) with each term inside the parentheses.",
       "latex": "2x \\cdot 3x = 6x^2, \\quad 2x \\cdot 4 = 8x"
     },
     {
       "text": "Combine the results: \\( 6x^2 + 8x \\).",
       "latex": "6x^2 + 8x"
     },
     {
       "text": "Now try multiplying two binomials: \\( (x + 2)(x + 3) \\). You can use the FOIL method.",
       "latex": "(x + 2)(x + 3)"
     },
     {
       "text": "FOIL stands for: First, Outer, Inner, Last.",
       "latex": ""
     },
     {
       "text": "First: \\(x \\cdot x = x^2 \\), Outer: \\(x \\cdot 3 = 3x\\), Inner: \\( 2 \\cdot x = 2x \\), Last: \\( 2 \\cdot 3 = 6 \\).",
       "latex": "x^2 + 3x + 2x + 6"
     },
     {
       "text": "Now combine like terms: \\( x^2 + 5x + 6 \\).",
       "latex": "x^2 + 5x + 6"
     },
     {
       "text": "Let’s go further with a trinomial and a binomial: \\( (x^2 + 2x + 1)(x + 4) \\).",
       "latex": "(x^2 + 2x + 1)(x + 4)"
     },
     {
       "text": "Distribute each term from the trinomial across the binomial:",
       "latex": "x^2(x + 4) = x^3 + 4x^2, \\quad 2x(x + 4) = 2x^2 + 8x, \\quad 1(x + 4) = x + 4"
     },
     {
       "text": "Now combine all the terms: \\( x^3 + 4x^2 + 2x^2 + 8x + x + 4 \\).",
       "latex": "x^3 + 4x^2 + 2x^2 + 8x + x + 4"
     },
     {
       "text": "Finally, simplify: \\(x^3 + 6x^2 + 9x + 4 \\).",
       "latex": "x^3 + 6x^2 + 9x + 4"
     },
     {
       "text": "Now let’s try multiplying two trinomials: \\( (x^2 + x + 1)(x^2 - x + 2) \\).",
       "latex": "(x^2 + x + 1)(x^2 - x + 2)"
     },
     {
       "text": "Distribute each term from the first trinomial across all terms in the second trinomial:",
       "latex": "x^2(x^2 - x + 2) = x^4 - x^3 + 2x^2, \\quad x(x^2 - x + 2) = x^3 - x^2 + 2x, \\quad 1(x^2 - x + 2) = x^2 - x + 2"
     },
     {
       "text": "Now combine all the terms: \\( x^4 - x^3 + 2x^2 + x^3 - x^2 + 2x + x^2 - x + 2 \\).",
       "latex": "x^4 - x^3 + 2x^2 + x^3 - x^2 + 2x + x^2 - x + 2"
     },
     {
       "text": "Simplify: \\( x^4 + 2x^2 + 2x + 2 \\).",
       "latex": "x^4 + 2x^2 + 2x + 2"
     }
   ],
   "outro": {
     "text": "To multiply polynomials, apply the distributive property — or FOIL for binomials — and combine like terms. With practice, you’ll multiply any two polynomials with confidence.",
     "audio": "To multiply polynomials, apply the distributive property — or FOIL for binomials — and combine like terms. With practice, you’ll multiply any two polynomials with confidence."
   }
 }
}

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