What value of [tex]$t$[/tex] makes the statement true?
[tex]$\begin{array}{l}
\left(4 x^3+5\right)\left(2 x^3+3\right)=8 x^6+t x^3+15 \\
t=
\end{array}$[/tex]
Answer (1)
Expand the left side of the equation: ( 4 x 3 + 5 ) ( 2 x 3 + 3 ) = 8 x 6 + 12 x 3 + 10 x 3 + 15 .
Combine like terms: 8 x 6 + 22 x 3 + 15 .
Compare the coefficients of x 3 on both sides of the equation.
The value of t is 22 .
Explanation
Understanding the Problem We are given the equation ( 4 x 3 + 5 ) ( 2 x 3 + 3 ) = 8 x 6 + t x 3 + 15 and we need to find the value of t .
Plan of Action To find the value of t , we need to expand the left side of the equation and then compare the coefficients of the x 3 term on both sides.
Expanding the Left Side Expanding the left side of the equation, we have: ( 4 x 3 + 5 ) ( 2 x 3 + 3 ) = 4 x 3 ( 2 x 3 ) + 4 x 3 ( 3 ) + 5 ( 2 x 3 ) + 5 ( 3 ) = 8 x 6 + 12 x 3 + 10 x 3 + 15 = 8 x 6 + 22 x 3 + 15
Comparing Coefficients Now, we compare the expanded form with the right side of the equation: 8 x 6 + 22 x 3 + 15 = 8 x 6 + t x 3 + 15
Finding the Value of t By comparing the coefficients of the x 3 terms, we can see that t = 22 .
Final Answer Therefore, the value of t that makes the statement true is 22.
Examples
Understanding polynomial expansion is crucial in various fields like physics, engineering, and computer graphics. For instance, when designing a bridge, engineers use polynomial equations to model the load distribution. By expanding these polynomials, they can accurately predict stress points and ensure the bridge's structural integrity. Similarly, in computer graphics, polynomial expansions are used to create smooth curves and surfaces, enhancing the visual realism of 3D models.
