Multiply and simplify.
$\left(7 n^3-\frac{28}{9} n^2-\frac{14}{5} n\right)\left(\frac{2}{7} n^2\right)$
$\left(7 n^3-\frac{28}{9} n^2-\frac{14}{5} n\right)\left(\frac{2}{7} n^2\right)=$
(Use integers or fractions for any number in the answer.)
Answer (1)
Distribute the term 7 2 n 2 to each term inside the parenthesis.
Multiply the coefficients and add the exponents of n for each term.
Simplify the resulting expression.
The final simplified expression is 2 n 5 − 9 8 n 4 − 5 4 n 3 .
Explanation
Understanding the Problem We are asked to multiply and simplify the expression ( 7 n 3 − 9 28 n 2 − 5 14 n ) ( 7 2 n 2 ) . This involves distributing the term 7 2 n 2 to each term inside the parentheses and then simplifying.
Multiplying the First Term First, distribute 7 2 n 2 to 7 n 3 :
7 2 n 2 × 7 n 3 = 7 2 × 7 × n 2 × n 3 = 2 n 2 + 3 = 2 n 5
Multiplying the Second Term Next, distribute 7 2 n 2 to − 9 28 n 2 :
7 2 n 2 × − 9 28 n 2 = 7 2 × − 9 28 × n 2 × n 2 = − 7 × 9 2 × 28 n 2 + 2 = − 63 56 n 4 = − 9 8 n 4
Multiplying the Third Term Finally, distribute 7 2 n 2 to − 5 14 n :
7 2 n 2 × − 5 14 n = 7 2 × − 5 14 × n 2 × n = − 7 × 5 2 × 14 n 2 + 1 = − 35 28 n 3 = − 5 4 n 3
Combining the Terms Now, combine the results: 2 n 5 − 9 8 n 4 − 5 4 n 3
Final Answer Therefore, the simplified expression is 2 n 5 − 9 8 n 4 − 5 4 n 3 .
Examples
Understanding polynomial multiplication is crucial in various fields, such as physics and engineering, where complex systems are modeled using polynomial equations. For example, when analyzing the trajectory of a projectile, engineers use polynomial functions to describe its path. Multiplying polynomials helps in combining different factors affecting the projectile's motion, such as initial velocity and gravitational forces, to predict its landing point accurately. This skill is also fundamental in computer graphics for rendering complex shapes and animations.
