Question

exercise 5
directions: choose the best answer from the four choices given. answers are on page 281.

1. what is the greatest common factor of $8x^3y^4z^6$, $12x^5y^3z^7$, and $24x^4yz^5$?

a. $2x^{11}y^8z^{18}$
b. $4x^3y^3z^6$
c. $24x^3yz^5$
d. $4x^3yz^5$

1. which of the following expressions is equivalent to $x^2 - y^2$?

a. $x^2 - 2xy - y^2$
b. $(x + y)(x + y)$
c. $(x - y)(x + y)$
d. $(x - y)(x - y)$

1. which of the following expressions is equivalent to $\frac{x^2 - x - 6}{x + 2}$?

a. $x^2 - \frac{x}{2} - 3$
b. $x^2 - 2$
c. $x - 2$
d. $x - 3$

1. if $x^2 + 30 = 11x$, then which of the following shows all possible values of $x$?

a. $\\{-5, 6\\}$
b. $\\{5, 6\\}$
c. $\\{5, -6\\}$
d. $\\{3, 10\\}$

1. which of the following is the solution set for $3x^2 + 3x = 6$?

a. $\\{1, -2\\}$
b. $\\{1, 2\\}$
c. $\left\\{\frac{1}{2}, 1\
ight\\}$
d. $\\{-1, -2\\}$

Explanation:

Question 1

Step1: Find GCF of coefficients

The coefficients are 8, 12, 24. Prime factors: \(8 = 2^3\), \(12 = 2^2\times3\), \(24 = 2^3\times3\). GCF of coefficients is \(2^2 = 4\).

Step2: Find GCF of \(x\)-terms

For \(x^3\), \(x^5\), \(x^4\), the lowest power is \(x^3\).

Step3: Find GCF of \(y\)-terms

For \(y^4\), \(y^3\), \(y^1\), the lowest power is \(y^1\) (i.e., \(y\)).

Step4: Find GCF of \(z\)-terms

For \(z^6\), \(z^7\), \(z^5\), the lowest power is \(z^5\).

Step5: Combine GCFs

Multiply the GCFs: \(4\times x^3\times y\times z^5 = 4x^3yz^5\)? Wait, no, wait for \(y\): \(y^4\), \(y^3\), \(y\) – wait, original terms: \(8x^3y^4z^6\), \(12x^5y^3z^7\), \(24x^4yz^5\). So \(y\) terms: \(y^4\), \(y^3\), \(y^1\). The lowest exponent is 1, so \(y^1\). \(z\) terms: \(z^6\), \(z^7\), \(z^5\) – lowest exponent 5. \(x\) terms: \(x^3\), \(x^5\), \(x^4\) – lowest exponent 3. Coefficient GCF: 4. Wait, but option D is \(4x^3yz^5\), but wait, let's check \(y\) in the second term: \(y^3\), third term \(y\). Wait, no, first term \(y^4\), second \(y^3\), third \(y\). So the GCF for \(y\) is \(y^1\) (since 1 is the lowest exponent). For \(x\): 3, for \(z\):5. Coefficient: 4. Wait, but option B is \(4x^3y^3z^6\) – no, that's higher. Wait, no, let's recalculate:

Wait, \(8x^3y^4z^6 = 2^3x^3y^4z^6\)

\(12x^5y^3z^7 = 2^2\times3x^5y^3z^7\)

\(24x^4yz^5 = 2^3\times3x^4yz^5\)

GCF of coefficients: \(2^2 = 4\)

GCF of \(x\): minimum exponent among 3,5,4 is 3: \(x^3\)

GCF of \(y\): minimum exponent among 4,3,1 is 1: \(y^1\)

GCF of \(z\): minimum exponent among 6,7,5 is 5: \(z^5\)

So GCF is \(4x^3yz^5\)? But option D is \(4x^3yz^5\), but wait, let's check the options again. Wait, option B is \(4x^3y^3z^6\) – no, that's not right. Wait, maybe I made a mistake. Wait, first term \(y^4\), second \(y^3\), third \(y\). So the GCF for \(y\) is \(y\) (since 1 is the lowest). For \(z\): 5. So the GCF is \(4x^3yz^5\), which is option D? Wait, no, wait the options:

A. \(2x^{11}y^8z^{18}\) – no, that's LCM.

B. \(4x^3y^3z^6\) – exponents for \(y\) and \(z\) are too high.

C. \(24x^3yz^5\) – coefficient is too high.

D. \(4x^3yz^5\) – yes, that's correct. Wait, but wait, maybe I messed up \(y\). Wait, first term \(y^4\), second \(y^3\), third \(y\). The GCF of \(y^4\), \(y^3\), \(y\) is \(y\) (since \(y = y^1\), and 1 is the lowest exponent). So yes, D.

Recall the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\). Here \(a = x\), \(b = y\), so \(x^2 - y^2 = (x - y)(x + y)\).

Step1: Factor the numerator

Factor \(x^2 - x - 6\). Find two numbers that multiply to -6 and add to -1: -3 and 2. So \(x^2 - x - 6 = (x - 3)(x + 2)\).

Step2: Simplify the fraction

The expression is \(\frac{(x - 3)(x + 2)}{x + 2}\). Cancel \(x + 2\) (assuming \(x
eq -2\)), so we get \(x - 3\).

Answer:

D. \(4x^3yz^5\)

Question 2