Verified Answer :

  • The problem requires the product of \(f(x) = 3x^2 + x - 12\) and \(g(x) = -3x - 15\).
  • Distribute each term of \(f(x)\) to every term in \(g(x)\) to obtain: \(-9x^3 - 45x^2 - 3x^2 - 15x + 36x + 180\).
  • Simplify by combining like terms (e.g., \(-45x^2 - 3x^2 = -48x^2\) and \(-15x + 36x = 21x\)).
  • The final result in standard form is \(\boxed{-9x^3 - 48x^2 + 21x + 180}\).

Explanation

  1. Analysis and Objective

To find the product of the two polynomials \(f(x) = 3x^2 + x - 12\) and \(g(x) = -3x - 15\), we perform the multiplication \(h(x) = f(x) \cdot g(x)\). Our objective is to determine the resulting polynomial in standard form, which involves distributing each term of the first polynomial across each term of the second and then combining like terms.

  1. Distribution and Expansion

We write the product as:
\[h(x) = (3x^2 + x - 12)(-3x - 15)\]
Applying the distributive property, we multiply each term of \(f(x)\) by each term of \(g(x)\):
\[h(x) = 3x^2(-3x) + 3x^2(-15) + x(-3x) + x(-15) + (-12)(-3x) + (-12)(-15)\]
Calculating each product:

  • \(3x^2 \cdot (-3x) = -9x^3\)
  • \(3x^2 \cdot (-15) = -45x^2\)
  • \(x \cdot (-3x) = -3x^2\)
  • \(x \cdot (-15) = -15x\)
  • \((-12) \cdot (-3x) = 36x\)
  • \((-12) \cdot (-15) = 180\)

Substituting these back into the expression:
\[h(x) = -9x^3 - 45x^2 - 3x^2 - 15x + 36x + 180\]

  1. Combining Terms and Final Form

Now we group the like terms together:

  • The cubic term: \(-9x^3\)
  • The quadratic terms: \(-45x^2 - 3x^2 = -48x^2\)
  • The linear terms: \(-15x + 36x = 21x\)
  • The constant term: \(180\)

Combining these, we get the polynomial in standard form:
\[h(x) = -9x^3 - 48x^2 + 21x + 180\]
The final answer is \(\boxed{-9x^3 - 48x^2 + 21x + 180}\).

Examples

Polynomial multiplication is frequently used in fields like physics and engineering to model complex systems. For instance, when designing a rectangular container where the dimensions are variables (like width \(w=x-5\), length \(l=2x+3\), and height \(h=x\)), finding the volume involves multiplying these expressions: \(V = x(x-5)(2x+3)\). Expanding this leads to a cubic polynomial, which allows engineers to determine the capacity of the container for any given value of \(x\) and find optimal dimensions using calculus.

Related Questions