Verified Answer :
- Simplify the first equation by combining like terms and isolating the missing polynomial \(P(x)\), which yields \(P(x) = 6x^2\).
- Simplify the second equation by distributing the negative sign and isolating the missing polynomial \(Q(b)\), which yields \(Q(b) = -6b^3\).
- Verify both results by substituting them back into their respective original equations to ensure equality holds.
- The missing terms are \(\boxed{6x^2}\) and \(\boxed{-6b^3}\).
Explanation
- Analysis of the Problem
We are given two equations containing an unknown polynomial term represented by the question mark (?). Our goal is to isolate and determine the correct polynomial expression for each case. The expressions provided, '? = x^2' and '? = x^3', are placeholder identifiers and should be ignored; we will solve for the actual missing terms algebraically.
The first equation is:
\[\left(-4 x^2 + 6 x + 3\right) + (8 x + 5 + P(x)) = 2 x^2 + 14 x + 8\]
where \(P(x)\) is the unknown polynomial.
The second equation is:
\[\left(-b^3 + 3 b^2 + 8\right) - \left(Q(b) - 5 b^2 - 9\right) = 5 b^3 + 8 b^2 + 17\]
where \(Q(b)\) is the unknown polynomial.
- Solving for the First Missing Term
To find \(P(x)\) in the first equation, we start by simplifying the left-hand side.
Grouping the terms on the left:
\[(-4x^2 + 6x + 8x + 3 + 5) + P(x) = 2x^2 + 14x + 8\]
\[-4x^2 + 14x + 8 + P(x) = 2x^2 + 14x + 8\]
Next, we isolate \(P(x)\) by subtracting \((-4x^2 + 14x + 8)\) from both sides:
\[P(x) = (2x^2 + 14x + 8) - (-4x^2 + 14x + 8)\]
\[P(x) = 2x^2 + 4x^2 + 14x - 14x + 8 - 8\]
\[P(x) = 6x^2\]
So, the missing term in the first equation is \(6x^2\).
- Solving for the Second Missing Term
For the second equation, we start by distributing the negative sign across the parenthesis:
\[(-b^3 + 3b^2 + 8) - Q(b) + 5b^2 + 9 = 5b^3 + 8b^2 + 17\]
Combine the known terms on the left-hand side:
\[-b^3 + (3b^2 + 5b^2) + (8 + 9) - Q(b) = 5b^3 + 8b^2 + 17\]
\[-b^3 + 8b^2 + 17 - Q(b) = 5b^3 + 8b^2 + 17\]
To solve for \(Q(b)\), we can subtract \((-b^3 + 8b^2 + 17)\) from both sides:
\[-Q(b) = (5b^3 + 8b^2 + 17) - (-b^3 + 8b^2 + 17)\]
\[-Q(b) = 5b^3 + b^3 + 8b^2 - 8b^2 + 17 - 17\]
\[-Q(b) = 6b^3\]
Multiplying by \(-1\), we find:
\[Q(b) = -6b^3\]
Thus, the missing term in the second equation is \(-6b^3\).
- Conclusion and Final Answer
We have determined that for the first equation, the missing term is \(6x^2\). For the second equation, the missing term is \(-6b^3\). Substituting these back into the original equations confirms their validity:
Equation 1: \((-4x^2 + 6x + 3) + (8x + 5 + 6x^2) = 2x^2 + 14x + 8\). This simplifies to \(2x^2 + 14x + 8 = 2x^2 + 14x + 8\), which is correct.
Equation 2: \((-b^3 + 3b^2 + 8) - (-6b^3 - 5b^2 - 9) = 5b^3 + 8b^2 + 17\). This simplifies to \(5b^3 + 8b^2 + 17 = 5b^3 + 8b^2 + 17\), which is also correct. The final results are \(6x^2\) and \(-6b^3\).
Examples
Polynomial addition and subtraction are essential in fields like physics and engineering for modeling systems. For example, if you are modeling the trajectory of a projectile (where motion is defined by polynomials), you might need to combine different forces acting on an object, each represented by a polynomial expression. Adding these polynomials allows engineers to find the resultant force or net position at any given time, ensuring structures and machines operate safely and as expected.