Question

for #1 - 7.
a) simplify each expression to an equivalent expression in lowest terms.
b) identify the value(s) of the variable that makes each expression undefined.

1. $\frac{5x - 35}{5x}$
2. $\frac{x^{2}+8x - 33}{x^{2}+5x - 24}$
3. $\frac{a^{2}-16}{2a - 8}$
4. $\frac{x^{2}-25}{x^{2}+8x + 15}$
5. $\frac{3m^{2}-27}{m^{2}+3m - 18}$

Explanation:

1. For the expression $\frac{5x - 35}{5x}$

Step1: Factor out the greatest - common factor in the numerator

Factor out 5 from $5x-35$ to get $5(x - 7)$. So the expression becomes $\frac{5(x - 7)}{5x}$.

Step2: Cancel out the common factor

Cancel out the common factor 5 in the numerator and denominator. The simplified expression is $\frac{x - 7}{x}$.
To find the value that makes the expression undefined, set the denominator equal to 0.
Set $5x=0$, then $x = 0$.

2. For the expression $\frac{x^{2}+8x - 33}{x^{2}+5x - 24}$

Step1: Factor the numerator and denominator

Factor the numerator $x^{2}+8x - 33=(x + 11)(x-3)$ using the formula $x^{2}+(a + b)x+ab=(x + a)(x + b)$ where $a = 11$ and $b=-3$.
Factor the denominator $x^{2}+5x - 24=(x + 8)(x-3)$.
So the expression is $\frac{(x + 11)(x - 3)}{(x + 8)(x - 3)}$.

Step2: Cancel out the common factor

Cancel out the common factor $(x - 3)$ (assuming $x
eq3$). The simplified expression is $\frac{x + 11}{x + 8}$.
Set the denominator $x^{2}+5x - 24=(x + 8)(x - 3)=0$. Then $x=-8$ or $x = 3$.

3. For the expression $\frac{a^{2}-16}{2a - 8}$

Step1: Factor the numerator and denominator

Factor the numerator $a^{2}-16=(a + 4)(a - 4)$ using the difference - of - squares formula $a^{2}-b^{2}=(a + b)(a - b)$ with $b = 4$.
Factor the denominator $2a-8=2(a - 4)$.
So the expression is $\frac{(a + 4)(a - 4)}{2(a - 4)}$.

Step2: Cancel out the common factor

Cancel out the common factor $(a - 4)$ (assuming $a
eq4$). The simplified expression is $\frac{a + 4}{2}$.
Set the denominator $2a-8 = 0$, then $2a=8$ and $a = 4$.

4. For the expression $\frac{x^{2}-25}{x^{2}+8x + 15}$

Answer:

1. a) $\frac{x - 7}{x}$; b) $x = 0$
2. a) $\frac{x + 11}{x + 8}$; b) $x=-8,x = 3$
3. a) $\frac{a + 4}{2}$; b) $a = 4$
4. a) $\frac{x - 5}{x + 3}$; b) $x=-3,x=-5$
5. a) $\frac{3m + 9}{m + 6}$; b) $m=-6,m = 3$