Answered

Which expression can be used to find the sum of the polynomials?

$\left(9-3 x^2

ight)+\left(-8 x^2+4 x+5

ight)$


A$\left[3 x^2+8 x^2

ight]+4 x+[9+(-5)]$

B$\left[\left(-3 x^2\right)+\left(-8 x^2\right)\right]+4 x+[9+5]$
C$\left[3 x^2+\left(-8 x^2\right)\right]+4 x+[9+5]$
D$\left[\left(-3 x^2\right)+\left(-8 x^2\right)\right]+4 x+[9+(-5)]$

Verified Answer :

  • Identify the like terms in the two polynomials \((9 - 3x^2)\) and \((-8x^2 + 4x + 5)\).
  • Regroup the terms by their power of \(x\): \((-3x^2 + (-8x^2))\) for \(x^2\), \(4x\) for \(x\), and \((9 + 5)\) for constants.
  • Construct the final expression: \([(-3x^2) + (-8x^2)] + 4x + [9 + 5]\).
  • Select the matching option, which is B.
  • Final Answer: \(\boxed{B}\)

Explanation

  1. Analyzing the Polynomials and Like Terms

The objective is to find the correct expression that represents the sum of the two given polynomials: \((9 - 3x^2)\) and \((-8x^2 + 4x + 5)\). Adding polynomials involves combining "like terms," which are terms that contain the same variable raised to the same power. To add these polynomials, we first group the terms by their power of \(x\).

  1. Grouping Like Terms

We have the sum of two polynomials: \((9 - 3x^2) + (-8x^2 + 4x + 5)\). To add them, we use the commutative and associative properties of addition to reorganize the terms so that terms with \(x^2\), terms with \(x\), and constant terms are grouped together.

  • For \(x^2\) terms, we have \(-3x^2\) from the first polynomial and \(-8x^2\) from the second. Grouping these gives \((-3x^2) + (-8x^2)\).
  • For the \(x\) term, there is only one term, which is \(4x\).
  • For the constant terms, we have \(9\) from the first polynomial and \(5\) from the second. Grouping these gives \(9 + 5\).
  1. Identifying the Correct Option

Combining the grouped terms, the expression becomes:
\[[(-3x^2) + (-8x^2)] + 4x + [9 + 5]\]
Comparing this to the provided choices:

A\([3x^2+8x^2]+4x+[9+(-5)]\)
B\([(-3x^2)+(-8x^2)]+4x+[9+5]\)
C\([3x^2+(-8x^2)]+4x+[9+5]\)
D\([(-3x^2)+(-8x^2)]+4x+[9+(-5)]\)

Option B correctly reflects our regrouping process of the terms.

Examples

In construction, if you need to calculate the total amount of materials for two different phases of a project, you often sum algebraic expressions representing those quantities. For example, if the first phase requires \(-3x^2 + 9\) units of a resource and the second phase requires \(-8x^2 + 4x + 5\) units, adding these expressions using the grouping method ensures that all like resources are tallied together to provide the correct total, \(-11x^2 + 4x + 14\). This helps in accurate resource planning and ordering.

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