What value of $t$ makes the statement true?
$\left(4 x^3+5\right)\left(2 x^3+3\right)=8 x^6+t x^3+15$
$t=$
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 .
Simplify the expanded form: 8 x 6 + 22 x 3 + 15 .
Compare the coefficients of x 3 on both sides of the equation: t = 22 .
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 asked to find the value of t that makes the equation true. To do this, we need to expand the left side of the equation and then compare the coefficients of the x 3 term with the right side.
Expanding the Left Side Let's expand the left side of the equation by multiplying the two binomials: ( 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 8 x 6 + 22 x 3 + 15 with the right side of the equation 8 x 6 + t x 3 + 15 . We can see that the coefficient of the x 3 term on the left side is 22, and on the right side, it is t . Therefore, we must have t = 22 .
Finding the Value of t Thus, the value of t that makes the statement true is 22.
Examples
Understanding polynomial multiplication is crucial in various fields, such as engineering and computer science. For example, when designing filters for signal processing, engineers often need to multiply polynomials to analyze the filter's frequency response. Similarly, in computer graphics, polynomial multiplication is used to perform transformations and rendering operations. By mastering polynomial multiplication, students can develop a strong foundation for tackling more complex problems in these fields.
