Question

to answer questions 7 - 9.

1. karmen factored (-18x^2 - 27x) but part of her solution was erased. find the missing part of her solution.

(-9x(square + 3))

1. kate and chet are factoring the polynomials shown below. they claim that their polynomials have the same gcf. do you agree or disagree? then, factor each of their polynomials.

kate
(4x^2 - 8x + 40)
chet
(20x^2 - 24x)
agree or disagree? explain:
kate: __________ chet: __________

1. an adventure crew takes a hike in the woods. the path of their hike is shown below. write an expression to represent the total distance of the hike and then factor the expression.

(4x + 1)
(5x + 4)
(6x + 5)
total distance: ________
factor: ________
©maneuvering the middle llc 2020

Explanation:

Question 7

Step 1: Recall factoring by GCF

To factor \(-18x^2 - 27x\), we first find the greatest common factor (GCF) of the two terms. The GCF of \(-18x^2\) and \(-27x\) is \(-9x\).

Step 2: Divide each term by the GCF

Divide \(-18x^2\) by \(-9x\): \(\frac{-18x^2}{-9x} = 2x\). Divide \(-27x\) by \(-9x\): \(\frac{-27x}{-9x} = 3\). So the factored form is \(-9x(2x + 3)\).

• GCF Calculation: For Kate's polynomial \(4x^2 - 8x + 40\), the GCF of \(4x^2\), \(-8x\), and \(40\) is \(4\) (since \(4\) is the largest number that divides \(4\), \(8\), and \(40\), and there's no common \(x\) term in all three). For Chet's polynomial \(20x^2 - 24x\), the GCF of \(20x^2\) and \(-24x\) is \(4x\) (GCF of \(20\) and \(24\) is \(4\), and GCF of \(x^2\) and \(x\) is \(x\)). Since \(4

eq 4x\), their polynomials do not have the same GCF.

• Factoring Kate's Polynomial: Factor out \(4\) from \(4x^2 - 8x + 40\): \(4(x^2 - 2x + 10)\).
• Factoring Chet's Polynomial: Factor out \(4x\) from \(20x^2 - 24x\): \(4x(5x - 6)\).

Step 1: Find total distance

To find the total distance of the hike, we add the lengths of the three sides: \((4x + 1) + (5x + 4) + (6x + 5)\).

Step 2: Combine like terms

Combine the \(x\)-terms: \(4x + 5x + 6x = 15x\). Combine the constant terms: \(1 + 4 + 5 = 10\). So the total distance is \(15x + 10\).

Step 3: Factor the expression

The GCF of \(15x\) and \(10\) is \(5\). Factor out \(5\): \(5(3x + 2)\).

Answer:

The missing part is \(2x\).

Question 8