Question

factor the trinomial. explain what tool or technique you used.

1. ( x^2 - 4x - 5 )
2. ( t^2 + t - 20 )
3. ( z^2 - z - 72 )

factor each expression. check your answer.

1. ( m^2 - 6m - 27 )
2. ( a^2 + 4a - 21 )
3. ( v^2 - 4v - 12 )
4. ( c^2 - 7c - 44 )
5. ( r^2 + 6r - 16 )
6. ( f^2 + f - 6 )
7. ( j^2 - 6j - 55 )
8. ( y^2 + 3y - 54 )
9. ( n^2 - 10n - 11 )

1. the area of a rectangular window is given by the trinomial ( x^2 - 14x + 48 ). the window’s length is ( (x - 8) ). what is the window’s width?

1. the area of a rectangular area rug is given by the trinomial ( f^2 - 4f - 77 ). the length of the rug is ( (f + 7) ). what is the width of the rug?

1. reasoning write possible expressions for the length and the width of a rectangle with area ( x^2 + 13x + 42 ).

1. a rectangular tabletop has an area of ( t^2 + 2t - 99 ). what are possible dimensions of the tabletop? use factoring.

Explanation:

Step1: Find pair summing to -4, product -5

We need two numbers that add to $-4$ and multiply to $-5$: $-5$ and $1$.

Step2: Factor the trinomial

$x^2 - 4x -5=(x-5)(x+1)$

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Step1: Find pair summing to 1, product -20

We need two numbers that add to $1$ and multiply to $-20$: $5$ and $-4$.

Step2: Factor the trinomial

$t^2 + t -20=(t+5)(t-4)$

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Step1: Find pair summing to -1, product -72

We need two numbers that add to $-1$ and multiply to $-72$: $8$ and $-9$.

Step2: Factor the trinomial

$z^2 - z -72=(z+8)(z-9)$

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Step1: Find pair summing to -6, product -27

We need two numbers that add to $-6$ and multiply to $-27$: $3$ and $-9$.

Step2: Factor the trinomial

$m^2 -6m -27=(m+3)(m-9)$

Step3: Verify by expanding

$(m+3)(m-9)=m^2-9m+3m-27=m^2-6m-27$

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Step1: Find pair summing to 4, product -21

We need two numbers that add to $4$ and multiply to $-21$: $7$ and $-3$.

Step2: Factor the trinomial

$a^2 +4a -21=(a+7)(a-3)$

Step3: Verify by expanding

$(a+7)(a-3)=a^2-3a+7a-21=a^2+4a-21$

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Step1: Find pair summing to -4, product -12

We need two numbers that add to $-4$ and multiply to $-12$: $2$ and $-6$.

Step2: Factor the trinomial

$v^2 -4v -12=(v+2)(v-6)$

Step3: Verify by expanding

$(v+2)(v-6)=v^2-6v+2v-12=v^2-4v-12$

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Step1: Find pair summing to -7, product -44

We need two numbers that add to $-7$ and multiply to $-44$: $4$ and $-11$.

Step2: Factor the trinomial

$c^2 -7c -44=(c+4)(c-11)$

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Step1: Find pair summing to 6, product -16

We need two numbers that add to $6$ and multiply to $-16$: $8$ and $-2$.

Step2: Factor the trinomial

$r^2 +6r -16=(r+8)(r-2)$

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Step1: Find pair summing to 1, product -6

We need two numbers that add to $1$ and multiply to $-6$: $3$ and $-2$.

Step2: Factor the trinomial

$f^2 +f -6=(f+3)(f-2)$

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Step1: Find pair summing to -6, product -55

We need two numbers that add to $-6$ and multiply to $-55$: $5$ and $-11$.

Step2: Factor the trinomial

$j^2 -6j -55=(j+5)(j-11)$

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Step1: Find pair summing to 3, product -54

We need two numbers that add to $3$ and multiply to $-54$: $9$ and $-6$.

Step2: Factor the trinomial

$y^2 +3y -54=(y+9)(y-6)$

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Step1: Find pair summing to -10, product -11

We need two numbers that add to $-10$ and multiply to $-11$: $1$ and $-11$.

Step2: Factor the trinomial

$n^2 -10n -11=(n+1)(n-11)$

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Step1: Factor the area trinomial

Area $=x^2-14x+48$. Find two numbers adding to $-14$, multiplying to $48$: $-6$ and $-8$.
$x^2-14x+48=(x-6)(x-8)$

Step2: Identify width

Since area = length × width, and length $=(x-8)$, width is $(x-6)$.

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Step1: Factor the area trinomial

Area $=f^2-4f-77$. Find two numbers adding to $-4$, multiplying to $-77$: $7$ and $-11$.
$f^2-4f-77=(f+7)(f-11)$

Step2: Identify width

Since area = length × width, and length $=(f+7)$, width is $(f-11)$.

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Step1: Factor the area trinomial

Area $=x^2+13x+42$. Find two numbers adding to $13$, multiplying to $42$: $6$ and $7$.
$x^2+13x+42=(x+6)(x+7)$

Step2: State length/width

Length and width are $(x+6)$ and $(x+7)$ (order does not matter).

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Step1: Factor the area trinomial

Area $=t^2+2t-99$. Find two numbers adding to $2$, multiplying to $-99$: $11$ and $-9$.
$t^2+2t-99=(t+11)(t-9)$

Step2: State possible dimensions

Possible dimensions are $(t+11)$ and $(t-9)$ (order does not matter).

Answer:

1. $(x-5)(x+1)$
2. $(t+5)(t-4)$
3. $(z+8)(z-9)$
4. $(m+3)(m-9)$
5. $(a+7)(a-3)$
6. $(v+2)(v-6)$
7. $(c+4)(c-11)$
8. $(r+8)(r-2)$
9. $(f+3)(f-2)$
10. $(j+5)(j-11)$
11. $(y+9)(y-6)$
12. $(n+1)(n-11)$
13. $x-6$
14. $f-11$
15. Length: $x+6$, Width: $x+7$ (or vice versa)
16. Length: $t+11$, Width: $t-9$ (or vice versa)