Sara subtracted two polynomials. She asked a friend to check her solution and work. What should her friend tell her? Is Sara correct? If she is not correct, explain which step the error was made AND what the error was.

Question

(2x3−7x2+4x)−(9x−5x3+9x2)
(
2
𝑥
3
−
7
𝑥
2
+
4
𝑥
)
−
(
9
𝑥
−
5
𝑥
3
+
9
𝑥
2
)


Step 1: 2x3−7x2+4x−9x−5x3+9x2
2
𝑥
3
−
7
𝑥
2
+
4
𝑥
−
9
𝑥
−
5
𝑥
3
+
9
𝑥
2


Step 2: 2x3−5x3−7x2+9x2+4x−9x
2
𝑥
3
−
5
𝑥
3
−
7
𝑥
2
+
9
𝑥
2
+
4
𝑥
−
9
𝑥


Step 3: −3x3+2x2−5x
−
3
𝑥
3
+
2
𝑥
2
−
5
𝑥


(2 points)

Answer 1

Sara's calculations contain an error. Let's go through the steps she provided and identify the mistake.

Step 1: In this step, Sara correctly set up the expression by rewriting the subtraction of the two polynomials:

\[ 2x^3 - 7x^2 + 4x - 9x + 5x^3 - 9x^2 \]

Step 2: In this step, she correctly combined like terms, but the order in which she did it is a bit confusing:

\[ 2x^3 - 5x^3 - 7x^2 + 9x^2 + 4x - 9x \]

It looks like she made an error with the signs when she wrote \(-5x^3\). In fact, when subtracting \(-5x^3\), it becomes \(+5x^3\), so it should be:

\[ 2x^3 + 5x^3 - 7x^2 + 9x^2 + 4x - 9x \]

Step 3: From her expression in Step 2, Sara combines the coefficients incorrectly. The correct combinations should be:

• For \(x^3\): \(2x^3 + 5x^3 = 7x^3\)
• For \(x^2\): \(-7x^2 + 9x^2 = 2x^2\)
• For \(x\): \(4x - 9x = -5x\)

Putting all together correctly would yield:

\[ 7x^3 + 2x^2 - 5x \]

Conclusion

Sara made her error in Step 2 by misapplying the subtraction of \(-5x^3\) and also in combining like terms. The final answer should be:

\[ 7x^3 + 2x^2 - 5x \]

Her friend should tell her that she did not accurately carry out the subtraction and therefore should recalculate the result.