Question

1. check for reasonableness factor -24a³b³ - 36a³b⁴. how can you check your work?
2. higher order thinking factor the expression $\frac{1}{12}x^{4}y^{3}-\frac{1}{9}x^{5}y^{2}$.
3. completely factor the expression $m^{8}n^{4}p^{5}-m^{2}n^{6}p^{3}$.
4. two students factor -15x⁷ - 60x⁶. which student is correct? explain.

student a -5x⁶(3x + 12)
student b 5x⁶(-3x + 12)

1. analyze and persevere the area of a rectangle is a = 36x² - 24x square inches. what are three different possible dimensions of the rectangle?
2. select the common factors for the expression -14x + 16x⁶.

2x
4x
-2x
-4x
2x²

1. select all of the expressions equivalent to -15w + 45w².

(-1 + 3w)15w
-15w(1 - 3w)
-15w(-1 - 3w)
15w(1 - 3w)
(-w + 3w²)15

Explanation:

Step1: Find GCF for -24a³b³ - 36a³b⁴

Find greatest - common - factor of coefficients and variables. Coefficients: GCF of 24 and 36 is 12. Variables: For a, lowest power is a³; for b, lowest power is b³. So GCF is - 12a³b³.

$$ - 24a^{3}b^{3}-36a^{3}b^{4}=-12a^{3}b^{3}(2 + 3b)$$

Step2: Check work

Multiply the factors. \(-12a^{3}b^{3}(2 + 3b)=-12a^{3}b^{3}\times2-12a^{3}b^{3}\times3b=-24a^{3}b^{3}-36a^{3}b^{4}\)

Step3: Factor \(\frac{1}{12}x^{4}y^{3}-\frac{1}{9}x^{5}y^{2}\)

Find GCF of coefficients and variables. Coefficients: GCF of \(\frac{1}{12}\) and \(\frac{1}{9}\) is \(\frac{1}{36}\). Variables: For x, lowest power is x⁴; for y, lowest power is y². So GCF is \(\frac{1}{36}x^{4}y^{2}\).

$$ \frac{1}{12}x^{4}y^{3}-\frac{1}{9}x^{5}y^{2}=\frac{1}{36}x^{4}y^{2}(3y - 4x)$$

Step4: Factor \(m^{8}n^{4}p^{5}-m^{2}n^{6}p^{3}\)

Find GCF of coefficients and variables. Coefficients: GCF is 1. Variables: For m, lowest power is m²; for n, lowest power is n⁴; for p, lowest power is p³. So GCF is \(m^{2}n^{4}p^{3}\).

$$m^{8}n^{4}p^{5}-m^{2}n^{6}p^{3}=m^{2}n^{4}p^{3}(m^{6}p^{2}-n^{2})$$

Step5: Analyze students' factoring of \(-15x^{7}-60x^{6}\)

Factor out GCF. Coefficients: GCF of 15 and 60 is 15. Variables: For x, lowest power is x⁶. So GCF is - 15x⁶.
\(-15x^{7}-60x^{6}=-15x^{6}(x + 4)\)
Student A: \(-5x^{6}(3x + 12)=-15x^{7}-60x^{6}\), but can be further factored.
Student B: \(5x^{6}(-3x + 12)=-15x^{7}+60x^{6}\), incorrect. So Student A is correct but not completely factored.

Step6: Find dimensions of rectangle with area \(A = 36x^{2}-24x\)

Factor out GCF. Coefficients: GCF of 36 and 24 is 12. Variables: For x, lowest power is x. So GCF is 12x.

$$A = 36x^{2}-24x=12x(3x - 2)$$

Three possible dimensions: \((12x)\times(3x - 2)\), \((6x)\times(6x - 4)\), \((4x)\times(9x - 6)\)

Step7: Find common factors of \(-14x+16x^{6}\)

Find GCF of coefficients and variables. Coefficients: GCF of 14 and 16 is 2. Variables: For x, lowest power is x. So GCF is 2x or - 2x.
Common factors are 2x and - 2x.

Step8: Find equivalent expressions of \(-15w + 45w^{2}\)

Factor out GCF. Coefficients: GCF of 15 and 45 is 15. Variables: For w, lowest power is w. So GCF is - 15w or 15w.
\(-15w + 45w^{2}=-15w(1 - 3w)=( - 1+3w)15w\)
Equivalent expressions are \((-1 + 3w)15w\) and \(-15w(1 - 3w)\)

Answer:

1. Factored form: \(-12a^{3}b^{3}(2 + 3b)\), check by multiplying factors.
2. Factored form: \(\frac{1}{36}x^{4}y^{2}(3y - 4x)\)
3. Factored form: \(m^{2}n^{4}p^{3}(m^{6}p^{2}-n^{2})\)
4. Student A is correct but not completely factored.
5. Three possible dimensions: \((12x)\times(3x - 2)\), \((6x)\times(6x - 4)\), \((4x)\times(9x - 6)\)
6. 2x, - 2x
7. \((-1 + 3w)15w\), \(-15w(1 - 3w)\)