Answer:
4) [tex]\frac{x}{7\cdot x +x^{2}}[/tex] is equivalent to [tex]\frac{1}{7+x}[/tex] for all [tex]x \ne -7[/tex]. (Answer: A)
5) [tex]\frac{-14\cdot x^{3}}{x^{3}-5\cdot x^{4}}[/tex] is equivalent to [tex]-\frac{14}{1-5\cdot x}[/tex] for all [tex]x \ne \frac{1}{5}[/tex]. (Answer: B)
6) [tex]\frac{x+7}{x^{2}+4\cdot x - 21}[/tex] is equivalent to [tex]\frac{1}{x-3}[/tex] for all [tex]x \ne 3[/tex]. (Answer: None)
7) [tex]\frac{x^{2}+3\cdot x -4}{x+4}[/tex] is equivalent to [tex]x - 1[/tex]. (Answer: None)
8) [tex]\frac{2}{3\cdot a}\cdot \frac{2}{a^{2}}[/tex] is equivalent to [tex]\frac{4}{3\cdot a^{3}}[/tex] for all [tex]a\ne 0[/tex]. (Answer: A)
Step-by-step explanation:
We proceed to simplify each expression below:
4) [tex]\frac{x}{7\cdot x +x^{2}}[/tex]
(i) [tex]\frac{x}{7\cdot x +x^{2}}[/tex] Given
(ii) [tex]\frac{x}{x\cdot (7+x)}[/tex] Distributive property
(iii) [tex]\frac{1}{7+x} \cdot \frac{x}{x}[/tex] Distributive property
(iv) [tex]\frac{1}{7+x}[/tex] Existence of multiplicative inverse/Modulative property/Result
Rational functions are undefined when denominator equals 0. That is:
[tex]7+x = 0[/tex]
[tex]x = -7[/tex]
Hence, we conclude that [tex]\frac{x}{7\cdot x +x^{2}}[/tex] is equivalent to [tex]\frac{1}{7+x}[/tex] for all [tex]x \ne -7[/tex]. (Answer: A)
5) [tex]\frac{-14\cdot x^{3}}{x^{3}-5\cdot x^{4}}[/tex]
(i) [tex]\frac{-14\cdot x^{3}}{x^{3}-5\cdot x^{4}}[/tex] Given
(ii) [tex]\frac{x^{3}\cdot (-14)}{x^{3}\cdot (1-5\cdot x)}[/tex] Distributive property
(iii) [tex]\frac{x^{3}}{x^{3}} \cdot \left(-\frac{14}{1-5\cdot x} \right)[/tex] Distributive property
(iv) [tex]-\frac{14}{1-5\cdot x}[/tex] Commutative property/Existence of multiplicative inverse/Modulative property/Result
Rational functions are undefined when denominator equals 0. That is:
[tex]1-5\cdot x = 0[/tex]
[tex]5\cdot x = 1[/tex]
[tex]x = \frac{1}{5}[/tex]
Hence, we conclude that [tex]\frac{-14\cdot x^{3}}{x^{3}-5\cdot x^{4}}[/tex] is equivalent to [tex]-\frac{14}{1-5\cdot x}[/tex] for all [tex]x \ne \frac{1}{5}[/tex]. (Answer: B)
6) [tex]\frac{x+7}{x^{2}+4\cdot x - 21}[/tex]
(i) [tex]\frac{x+7}{x^{2}+4\cdot x - 21}[/tex] Given
(ii) [tex]\frac{x+7}{(x+7)\cdot (x-3)}[/tex] [tex]x^{2} -(r_{1}+r_{2})\cdot x +r_{1}\cdot r_{2} = (x-r_{1})\cdot (x-r_{2})[/tex]
(iii) [tex]\frac{1}{x-3}\cdot \frac{x+7}{x+7}[/tex] Commutative and distributive properties.
(iv) [tex]\frac{1}{x-3}[/tex] Existence of multiplicative inverse/Modulative property/Result
Rational functions are undefined when denominator equals 0. That is:
[tex]x-3 = 0[/tex]
[tex]x = 3[/tex]
Hence, we conclude that [tex]\frac{x+7}{x^{2}+4\cdot x - 21}[/tex] is equivalent to [tex]\frac{1}{x-3}[/tex] for all [tex]x \ne 3[/tex]. (Answer: None)
7) [tex]\frac{x^{2}+3\cdot x -4}{x+4}[/tex]
(i) [tex]\frac{x^{2}+3\cdot x -4}{x+4}[/tex] Given
(ii) [tex]\frac{(x+4)\cdot (x-1)}{x+4}[/tex] [tex]x^{2} -(r_{1}+r_{2})\cdot x +r_{1}\cdot r_{2} = (x-r_{1})\cdot (x-r_{2})[/tex]
(iii) [tex](x-1)\cdot \left(\frac{x+4}{x+4} \right)[/tex] Commutative and distributive properties.
(iv) [tex]x - 1[/tex] Existence of additive inverse/Modulative property/Result
Polynomic function are defined for all value of [tex]x[/tex].
[tex]\frac{x^{2}+3\cdot x -4}{x+4}[/tex] is equivalent to [tex]x - 1[/tex]. (Answer: None)
8) [tex]\frac{2}{3\cdot a}\cdot \frac{2}{a^{2}}[/tex]
(i) [tex]\frac{2}{3\cdot a}\cdot \frac{2}{a^{2}}[/tex]
(ii) [tex]\frac{4}{3\cdot a^{3}}[/tex] [tex]\frac{a}{b}\cdot \frac{c}{d} = \frac{a\cdot b}{c\cdot d}[/tex]/Result
Rational functions are undefined when denominator equals 0. That is:
[tex]3\cdot a^{3} = 0[/tex]
[tex]a = 0[/tex]
Hence, [tex]\frac{2}{3\cdot a}\cdot \frac{2}{a^{2}}[/tex] is equivalent to [tex]\frac{4}{3\cdot a^{3}}[/tex] for all [tex]a\ne 0[/tex]. (Answer: A)
