Question

13 this question has two parts.
students in an algebra class were multiplying binomials.
anna, beth, charles and dave were discussing the multiplication problem below:
multiply: (p - 4)(p + 2)
each student got a different answer.
anna claimed the answer was: ( p^2 - 2p - 8 )
beth claimed the answer was: ( p^2 - 4p - 2 )
charles claimed the answer was: ( p^2 + 8 )
dave claimed the answer was: ( p^2 - 2 )
part a: which students claim was correct?
part b: what evidence or reasoning could the student you chose in part a use to convince their classmates why their answer was correct?

14 markee subtracted the polynomials with the following steps. was he correct or not? explain your answer. ( (x^3 + 3x^2 - 6x + 2) - (3x^2 + x - 4) )

15 a rectangle has a width of ( x ) units. its length is 2 times the width, so the length is ( 2x ). the width is increased by 5 units, and the length is increased by 3 units. which expression represents the area of the new rectangle?
a) ( (x + 5)(2x + 3) )
b) ( (x + 3)(2x + 5) )
c) ( 2x^2 + 8x + 15 )
d) ( 2x^2 + 13x + 15 )

Explanation:

Part A

Step1: Multiply the binomials

To multiply \((p - 4)(p + 2)\), we use the distributive property (FOIL method):
First: \(p\times p = p^{2}\)
Outer: \(p\times2 = 2p\)
Inner: \(-4\times p=-4p\)
Last: \(-4\times2 = -8\)

Step2: Combine like terms

Combine the outer and inner terms: \(2p-4p=-2p\)
So the product is \(p^{2}-2p - 8\), which matches Anne's answer.

To convince classmates, the student (Anne) can use the distributive property (FOIL method). First, multiply the First terms \(p\times p = p^{2}\), then the Outer terms \(p\times2 = 2p\), then the Inner terms \(-4\times p=-4p\), and finally the Last terms \(-4\times2=-8\). Then combine the like - terms \(2p-4p=-2p\). So the result is \(p^{2}-2p - 8\), which shows that her answer is correct.

Step1: Recall polynomial subtraction rule

When subtracting polynomials \((a + b)-(c + d)=a + b - c - d\), we need to distribute the negative sign to each term in the second polynomial.
Given \((x^{3}+2x^{2}-6x + 2)-(3x^{2}+x - 4)\), we should get \(x^{3}+2x^{2}-6x + 2-3x^{2}-x + 4\)

Step2: Check Markee's steps (from the visible work)

From the visible work, it seems Markee did not distribute the negative sign correctly to all terms (he might have missed changing the sign of \(-4\) to \(+ 4\) and the sign of \(x\) to \(-x\) properly). So his steps are incorrect.

Answer:

Anne's claim was correct.

Part B