Algebra 2 Practice Worksheet | Worksheets Worksheets - Free Printable

Here are the step-by-step solutions for the problems on your worksheet.

Part 1: Find all zeros

To find the zeros of a function, you set each factor equal to zero and solve for $x$. This works because if any part of a multiplication problem is zero, the whole answer is zero.

1. $f(x) = (x-5)(2x+1)$
* Set the first part to 0: $x - 5 = 0 \rightarrow x = 5$
* Set the second part to 0: $2x + 1 = 0 \rightarrow 2x = -1 \rightarrow x = -\frac{1}{2}$
* Zeros: $5, -\frac{1}{2}$

2. $f(x) = (x-4)(x-3)$
* Set the first part to 0: $x - 4 = 0 \rightarrow x = 4$
* Set the second part to 0: $x - 3 = 0 \rightarrow x = 3$
* Zeros: $4, 3$

3. $f(x) = (2x-2)(x+5)$
* Set the first part to 0: $2x - 2 = 0 \rightarrow 2x = 2 \rightarrow x = 1$
* Set the second part to 0: $x + 5 = 0 \rightarrow x = -5$
* Zeros: $1, -5$

4. $f(x) = (3x-1)(x+9)(4x-1)$
* Set the first part to 0: $3x - 1 = 0 \rightarrow 3x = 1 \rightarrow x = \frac{1}{3}$
* Set the second part to 0: $x + 9 = 0 \rightarrow x = -9$
* Set the third part to 0: $4x - 1 = 0 \rightarrow 4x = 1 \rightarrow x = \frac{1}{4}$
* Zeros: $\frac{1}{3}, -9, \frac{1}{4}$

5. $f(x) = (x-8)(4x+2)(3x-8)$
* Set the first part to 0: $x - 8 = 0 \rightarrow x = 8$
* Set the second part to 0: $4x + 2 = 0 \rightarrow 4x = -2 \rightarrow x = -\frac{2}{4} = -\frac{1}{2}$
* Set the third part to 0: $3x - 8 = 0 \rightarrow 3x = 8 \rightarrow x = \frac{8}{3}$
* Zeros: $8, -\frac{1}{2}, \frac{8}{3}$

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Part 2: State the excluded values

Excluded values are numbers that make the denominator (the bottom number) of a fraction equal to zero. You cannot divide by zero. To find them, set the original denominator equal to zero and solve. Note: We look at the denominator *before* simplifying the expression.

6. $\frac{30x^3}{15x}$
* Look at the denominator: $15x$
* Set it to 0: $15x = 0$
* Solve: $x = 0$
* Excluded Value: $x \neq 0$
*(Simplified form: $2x^2$)*

7. $\frac{x^2 + 8x + 7}{x + 1}$
* Look at the denominator: $x + 1$
* Set it to 0: $x + 1 = 0$
* Solve: $x = -1$
* Excluded Value: $x \neq -1$
*(Simplified form: $x + 7$)*

8. $\frac{60x^2}{40x}$
* Look at the denominator: $40x$
* Set it to 0: $40x = 0$
* Solve: $x = 0$
* Excluded Value: $x \neq 0$
*(Simplified form: $\frac{3x}{2}$ or $1.5x$)*

Final Answer:
1. $x = 5, -\frac{1}{2}$
2. $x = 4, 3$
3. $x = 1, -5$
4. $x = \frac{1}{3}, -9, \frac{1}{4}$
5. $x = 8, -\frac{1}{2}, \frac{8}{3}$
6. Excluded: $x \neq 0$; Simplified: $2x^2$
7. Excluded: $x \neq -1$; Simplified: $x + 7$
8. Excluded: $x \neq 0$; Simplified: $\frac{3x}{2}$