Question

describe the end behavior of each function.

1. $f(x)=x^2 - 8x + 15$ 2) $f(x)=-x^3 + 4x^2 - 7$

simplify each expression.

1. $(12r + 10r^2 + 9r^5)+(r + 6r^2)$ 4) $(14x^2 - 14x^4 - 11x)-(13x^2 + 4x^4)$

find each product.

1. $(2 - 3x)^2$ 6) $(2x - 6)(2x + 6)$
2. $(8a^2 + 8a - 1)(8a^2 + 6a + 6)$ 8) $(-3n^2 - 2n - 8)(-6n^2 + 2n - 4)$

expand completely.

1. $(y + x)^4$ 10) $(y - 3x)^3$

evaluate each function using synthetic division at the given value.

1. $f(n)=-5n^4 - 28n^3 + 13n^2 + 7n - 2$ at $n = -6$
2. $f(n)=n^4 + 3n^3 + 5n^2 + 5n - 12$ at $n = -2$

factor each completely.

1. $r^2 - 4$ 14) $27n^2 + 18n + 3$
2. $6r^3 - 5r^2 - 12r + 10$ 16) $4b^3 - 4b^2 - 360b$
3. $6a^3 - 30a^2 + 36a$ 18) $x^3 + 13x^2 + 36x$

divide. use long division use synthetic

1. $(x^3 - 10x^2 + 7x)÷(x - 1)$ 20) $(n^3 - 4n^2 + 9)÷(n - 4)$

Explanation:

---

1) End Behavior of $f(x)=x^2-8x+15$

Step1: Identify degree and leading coeff.

Degree = 2 (even), Leading coefficient = 1 (positive)

Step2: Define end behavior

As $x\to+\infty$, $f(x)\to+\infty$; As $x\to-\infty$, $f(x)\to+\infty$

---

2) End Behavior of $f(x)=-x^3+4x^2-7$

Step1: Identify degree and leading coeff.

Degree = 3 (odd), Leading coefficient = -1 (negative)

Step2: Define end behavior

As $x\to+\infty$, $f(x)\to-\infty$; As $x\to-\infty$, $f(x)\to+\infty$

---

3) Simplify $(12r+10r^2+9r^5)+(r+6r^2)$

Step1: Combine like terms

$9r^5 + (10r^2+6r^2) + (12r+r)$

Step2: Calculate sums

$9r^5 + 16r^2 + 13r$

---

4) Simplify $(14x^2-14x^4-11x)-(13x^2+4x^4)$

Step1: Distribute negative sign

$14x^2-14x^4-11x-13x^2-4x^4$

Step2: Combine like terms

$(-14x^4-4x^4)+(14x^2-13x^2)-11x$

Step3: Calculate sums

$-18x^4 + x^2 - 11x$

---

5) Find product $(2-3x)^2$

Step1: Apply square of difference

$(a-b)^2=a^2-2ab+b^2$, $a=2, b=3x$

Step2: Expand and simplify

$2^2 - 2(2)(3x) + (3x)^2 = 4 - 12x + 9x^2$

---

6) Find product $(2x-6)(2x+6)$

Step1: Apply difference of squares

$(a-b)(a+b)=a^2-b^2$, $a=2x, b=6$

Step2: Expand and simplify

$(2x)^2 - 6^2 = 4x^2 - 36$

---

7) Find product $(8a^2+8a-1)(8a^2+6a+6)$

Step1: Distribute each term

$8a^2(8a^2+6a+6)+8a(8a^2+6a+6)-1(8a^2+6a+6)$

Step2: Expand each term

$64a^4+48a^3+48a^2+64a^3+48a^2+48a-8a^2-6a-6$

Step3: Combine like terms

$64a^4+(48a^3+64a^3)+(48a^2+48a^2-8a^2)+(48a-6a)-6$

Step4: Calculate sums

$64a^4 + 112a^3 + 88a^2 + 42a - 6$

---

8) Find product $(-3n^2-2n-8)(-6n^2+2n-4)$

Step1: Distribute each term

$-3n^2(-6n^2+2n-4)-2n(-6n^2+2n-4)-8(-6n^2+2n-4)$

Step2: Expand each term

$18n^4-6n^3+12n^2+12n^3-4n^2+8n+48n^2-16n+32$

Step3: Combine like terms

$18n^4+(-6n^3+12n^3)+(12n^2-4n^2+48n^2)+(8n-16n)+32$

Step4: Calculate sums

$18n^4 + 6n^3 + 56n^2 - 8n + 32$

---

9) Expand $(y+x)^4$

Step1: Apply binomial theorem

$(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$, $a=y, b=x$

Step2: Substitute and expand

$y^4 + 4y^3x + 6y^2x^2 + 4yx^3 + x^4$

---

10) Expand $(y-3x)^3$

Step1: Apply binomial theorem

$(a-b)^3=a^3-3a^2b+3ab^2-b^3$, $a=y, b=3x$

Step2: Substitute and expand

$y^3 - 3y^2(3x) + 3y(3x)^2 - (3x)^3$

Step3: Simplify terms

$y^3 - 9y^2x + 27yx^2 - 27x^3$

---

11) Evaluate $f(n)=-5n^4-28n^3+13n^2+7n-2$ at $n=-6$ (Synthetic Division)

Step1: List coefficients and root

Coefficients: $-5, -28, 13, 7, -2$; Root: $-6$

Step2: Perform synthetic division

```
-6 | -5 -28 13 7 -2
30 -12 -6 -6
------------------------
-5 2 1 1 -8
```

Step3: Identify remainder (function value)

Remainder = $-8$

---

12) Evaluate $f(n)=n^4+3n^3+5n^2+5n-12$ at $n=-2$ (Synthetic Division)

Step1: List coefficients and root

Coefficients: $1, 3, 5, 5, -12$; Root: $-2$

Step2: Perform synthetic division

```
-2 | 1 3 5 5 -12
-2 -2 -6 2
-------------------
1 1 3 -1 -10
```

Step3: Identify remainder (function value)

Remainder = $-10$

---

13) Factor $r^2-4$

Step1: Apply difference of squares

$a^2-b^2=(a-b)(a+b)$, $a=r, b=2$

Step2: Write factored form

$(r-2)(r+2)$

---

14) Factor $27n^2+18n+3$

Step1: Factor out GCF (3)

$3(9n^2+6n+1)$

Step2: Factor quadratic

$9n^2+6n+1=(3n+1)^2$

Step3: Write final form

$3(3n+1)^2$

---

15) Factor $6r^3-5r^2-12r+10$

Step1: Group terms

$(6r^3-5r^2)+(-12r+10)$

Step2: Factor GCF from groups

$r^2(6r-5)-2(6r-5)$

Step3: Factor common binomial

$(r^2-2)(6r-5)$

---

16) Factor $4b^3-4b^2-360b$

Step1: Factor out GCF (4b)

$4b(b^2-b-90)$

Step2: Factor quadratic

$b^2-b-90=(b-10)(b+9)$

Step3: Write final form

$4b(b-10)(b+9)$

---

17) Factor $6a^3-30a^2+36a$

Step1: Factor out GCF (6a)

$6a(a^2-5a+6)$

Step2: Factor quadratic

$a^2-5a+6=(a-2)(a-3)$

Step3: Write final form

$6a(a-2)(a-3)$

---

18) Factor $x^3+13x^2+36x$

Step1: Factor out GCF (x)

$x(x^2+13x+36)$

Step2: Factor quadratic

$x^2+13x+36=(x+4)(x+9)$

Step3: Write final form

$x(x+4)(x+9)$

---

19) Divide $(x^3-10x^2+7x)\div(x-1)$ (Long Division)

Step1: Divide leading terms

$\frac{x^3}{x}=x^2$, multiply divisor: $x^2(x-1)=x^3-x^2$

Step2: Subtract from dividend

$(x^3-10x^2+7x)-(x^3-x^2)=-9x^2+7x$

Step3: Repeat division

$\frac{-9x^2}{x}=-9x$, multiply divisor: $-9x(x-1)=-9x^2+9x$

Step4: Subtract

$(-9x^2+7x)-(-9x^2+9x)=-2x$

Step5: Repeat division

$\frac{-2x}{x}=-2$, multiply divisor: $-2(x-1)=-2x+2$

Step6: Subtract to get remainder

$(-2x)-(-2x+2)=-2$

Step7: Write quotient + remainder

$x^2-9x-2 + \frac{-2}{x-1}$

---

20) Divide $(n^3-4n^2+9)\div(n-4)$ (Synthetic Division)

Step1: List coefficients and root

Coefficients: $1, -4, 0, 9$; Root: $4$

Step2: Perform synthetic division

```
4 | 1 -4 0 9
4 0 0
-------------
1 0 0 9
```

Step3: Write quotient + remainder

$n^2+0n+0 + \frac{9}{n-4}=n^2 + \frac{9}{n-4}$

---

Answer:

1. As $x\to+\infty$, $f(x)\to+\infty$; As $x\to-\infty$, $f(x)\to+\infty$
2. As $x\to+\infty$, $f(x)\to-\infty$; As $x\to-\infty$, $f(x)\to+\infty$
3. $9r^5 + 16r^2 + 13r$
4. $-18x^4 + x^2 - 11x$
5. $9x^2 - 12x + 4$
6. $4x^2 - 36$
7. $64a^4 + 112a^3 + 88a^2 + 42a - 6$
8. $18n^4 + 6n^3 + 56n^2 - 8n + 32$
9. $y^4 + 4xy^3 + 6x^2y^2 + 4x^3y + x^4$
10. $y^3 - 9xy^2 + 27x^2y - 27x^3$
11. $f(-6) = -8$
12. $f(-2) = -10$
13. $(r-2)(r+2)$
14. $3(3n+1)^2$
15. $(6r-5)(r^2-2)$
16. $4b(b-10)(b+9)$
17. $6a(a-2)(a-3)$
18. $x(x+4)(x+9)$
19. $x^2 - 9x - 2 - \frac{2}{x-1}$
20. $n^2 + \frac{9}{n-4}$