\left(-\frac{11}{2 \sqrt{2}} x\right) \times\left(-x^4+\frac{1}{\sqrt{2}} x^3-\frac{\sqrt{3}}{7} x^2+\frac{2}{5} x-\frac{21}{8}\right)
Answer (2)
Distribute the term − 2 2 11 x to each term in the polynomial.
Multiply and simplify each term:
( − 2 2 11 x ) × ( − x 4 ) = 2 2 11 x 5
( − 2 2 11 x ) × ( 2 1 x 3 ) = − 4 11 x 4
( − 2 2 11 x ) × ( − 7 3 x 2 ) = 14 2 11 3 x 3
( − 2 2 11 x ) × ( 5 2 x ) = − 5 2 11 x 2
( − 2 2 11 x ) × ( − 8 21 ) = 16 2 231 x
Combine the terms to form the final expression: 2 2 11 x 5 − 4 11 x 4 + 14 2 11 3 x 3 − 5 2 11 x 2 + 16 2 231 x
Explanation
Understanding the Problem We are given the expression ( − 2 2 11 x ) × ( − x 4 + 2 1 x 3 − 7 3 x 2 + 5 2 x − 8 21 ) Our goal is to expand and simplify this expression.
Solution Plan We need to distribute the linear term − 2 2 11 x to each term of the quartic polynomial − x 4 + 2 1 x 3 − 7 3 x 2 + 5 2 x − 8 21 .
Performing the Multiplication
Multiply − 2 2 11 x by − x 4 :
− 2 2 11 x × − x 4 = 2 2 11 x 5
Multiply − 2 2 11 x by 2 1 x 3 :
− 2 2 11 x × 2 1 x 3 = − 4 11 x 4
Multiply − 2 2 11 x by − 7 3 x 2 :
− 2 2 11 x × − 7 3 x 2 = 14 2 11 3 x 3
Multiply − 2 2 11 x by 5 2 x :
− 2 2 11 x × 5 2 x = − 5 2 11 x 2
Multiply − 2 2 11 x by − 8 21 :
− 2 2 11 x × − 8 21 = 16 2 231 x
Combining the Terms Now, we combine all the terms to get the expanded polynomial: 2 2 11 x 5 − 4 11 x 4 + 14 2 11 3 x 3 − 5 2 11 x 2 + 16 2 231 x We can approximate the coefficients: 2 2 11 ≈ 3.889 4 11 = 2.75 14 2 11 3 ≈ 0.962 5 2 11 ≈ 1.556 16 2 231 ≈ 10.209 So the polynomial is approximately: 3.889 x 5 − 2.75 x 4 + 0.962 x 3 − 1.556 x 2 + 10.209 x
Final Answer Thus, the final expanded and simplified expression is: 2 2 11 x 5 − 4 11 x 4 + 14 2 11 3 x 3 − 5 2 11 x 2 + 16 2 231 x
Examples
Polynomials are used to model curves and relationships in various fields. For example, engineers use polynomials to design bridges and predict the trajectory of projectiles. Economists use them to model growth and decay, and computer scientists use them in algorithms. Understanding how to manipulate and simplify polynomial expressions is crucial for solving real-world problems in these fields.
To expand the expression, we distribute − 2 2 11 x to each term in the polynomial, resulting in 2 2 11 x 5 − 4 11 x 4 + 14 2 11 3 x 3 − 5 2 11 x 2 + 16 2 231 x . Each multiplication step was performed by applying the rules of distributing terms among coefficients and variables. The final answer is a combination of simplified polynomial terms representing the distribution results.
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