Answered

Multiply:

$\left(2 x^2+4 x-3\right)\left(x^2-2 x+5\right)$


A$2 x^4-x^2+26 x-15$
B$2 x^4+8 x^3-x^2+26 x-15$
C$2 x^4-8 x^2-15$
D$2 x^4+7 x^2-15$

Verified Answer :

  • Distribute each term of the first polynomial across the second: \((2x^2 + 4x - 3)(x^2 - 2x + 5) = 2x^2(x^2 - 2x + 5) + 4x(x^2 - 2x + 5) - 3(x^2 - 2x + 5)\).
  • Expand and perform the multiplication for each component: \(2x^4 - 4x^3 + 10x^2 + 4x^3 - 8x^2 + 20x - 3x^2 + 6x - 15\).
  • Simplify by combining like terms: \(2x^4 + (-4x^3 + 4x^3) + (10x^2 - 8x^2 - 3x^2) + (20x + 6x) - 15\).
  • The final product is \(\boxed{2x^4 - x^2 + 26x - 15}\), which corresponds to option a.

Explanation

  1. Problem Analysis and Setup

To find the product of the two polynomials \((2x^2 + 4x - 3)\) and \((x^2 - 2x + 5)\), we apply the distributive property, which states that each term in the first polynomial must be multiplied by every term in the second polynomial. This expansion is essential for simplifying the expression into a standard polynomial form.

  1. Expanding the Polynomials

We expand the product by distributing each term of the first polynomial across the second:
\[(2x^2 + 4x - 3)(x^2 - 2x + 5) = 2x^2(x^2 - 2x + 5) + 4x(x^2 - 2x + 5) - 3(x^2 - 2x + 5).\]
Performing the multiplication for each term:

  1. \(2x^2(x^2 - 2x + 5) = 2x^4 - 4x^3 + 10x^2\)
  2. \(4x(x^2 - 2x + 5) = 4x^3 - 8x^2 + 20x\)
  3. \(-3(x^2 - 2x + 5) = -3x^2 + 6x - 15\)

Summing these results gives:
\[(2x^4 - 4x^3 + 10x^2) + (4x^3 - 8x^2 + 20x) - 3x^2 + 6x - 15.\]

  1. Combining Like Terms and Conclusion

Next, we group the like terms together to simplify the expression:
\[2x^4 + (-4x^3 + 4x^3) + (10x^2 - 8x^2 - 3x^2) + (20x + 6x) - 15.\]
Combining the coefficients for each power of \(x\):

  • The \(x^4\) term: \(2x^4\)
  • The \(x^3\) term: \(-4x^3 + 4x^3 = 0\)
  • The \(x^2\) term: \(10x^2 - 8x^2 - 3x^2 = -x^2\)
  • The \(x\) term: \(20x + 6x = 26x\)
  • The constant term: \(-15\)

Putting it all together, we get \(2x^4 - x^2 + 26x - 15\). This result matches option (a).

Examples

In engineering or physics, multiplying polynomials is essential for modeling complex systems, such as calculating the path of a projectile or determining the energy distribution in a structural beam. For example, if the velocity of an object is represented by one polynomial and its time duration by another, finding the product helps calculate total displacement or energy over time, allowing for more precise design and analysis in real-world engineering projects.