Question 3

Let's factor each expression using appropriate algebraic identities.

Step 1 — Factor $9a^2 + b^2 + 4c^2 - 6ab + 12ac - 4bc$

We see three squared terms and three product terms. This suggests the identity $(x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx$. Let's identify $x$, $y$, and $z$. We have $x^2 = 9a^2$, so $x = \mathbf{3a}$. We have $y^2 = b^2$. We have $z^2 = 4c^2$. Now, let's look at the product terms to find the signs. The term $-6ab$ is $2xy$. So, $2(3a)y = -6ab$. This means $6ay = -6ab$, so $y = \mathbf{-b}$. The term $12ac$ is $2xz$. So, $2(3a)z = 12ac$. This means $6az = 12ac$, so $z = \mathbf{2c}$. Let's check the last product term $2yz$. $2(-b)(2c) = -4bc$. This matches the given expression. So, the expression fits the identity with $x=3a$, $y=-b$, and $z=2c$.

$9a^2 + b^2 + 4c^2 - 6ab + 12ac - 4bc$

$= (3a)^2 + (-b)^2 + (2c)^2 + 2(3a)(-b) + 2(-b)(2c) + 2(3a)(2c)$

$\boxed{(3a - b + 2c)^2}$

Step 2 — Factor $16s^2 + 25t^2 - 40st$

This expression has two squared terms and one product term. This looks like the identity $(a-b)^2 = a^2 - 2ab + b^2$. Let's identify $a$ and $b$. We have $a^2 = 16s^2$, so $a = \mathbf{4s}$. We have $b^2 = 25t^2$, so $b = \mathbf{5t}$. Now, let's check the middle term. The term $-40st$ should be $-2ab$. So, $-2(4s)(5t) = -40st$. This matches the given expression. So, the expression fits the identity with $a=4s$ and $b=5t$.

$16s^2 + 25t^2 - 40st$

$= (4s)^2 - 2(4s)(5t) + (5t)^2$

$\boxed{(4s - 5t)^2}$

Step 3 — Factor $r^2 - r - 42$

This is a quadratic trinomial. We need to find two numbers. These numbers must multiply to give the constant term, which is $\mathbf{-42}$. They must also add up to give the coefficient of the middle term, which is $\mathbf{-1}$. Let's list factors of $\mathbf{42}$: $(1, 42)$, $(2, 21)$, $(3, 14)$, $(6, 7)$. Since the product is negative, one number is positive and one is negative. Since the sum is negative, the number with the larger absolute value must be negative. Let's try $\mathbf{-7}$ and $\mathbf{6}$. Their product is $(-7) \times 6 = \mathbf{-42}$. Their sum is $(-7) + 6 = \mathbf{-1}$. These are the correct numbers. Now, we rewrite the middle term using these numbers.

$r^2 - r - 42$

$= r^2 - 7r + 6r - 42$

$= r(r - 7) + 6(r - 7)$

$= (r - 7)(r + 6)$

$\boxed{(r - 7)(r + 6)}$

Step 4 — Factor $49g^2 + 14gh + h^2$

This expression has two squared terms and one product term. This looks like the identity $(a+b)^2 = a^2 + 2ab + b^2$. Let's identify $a$ and $b$. We have $a^2 = 49g^2$, so $a = \mathbf{7g}$. We have $b^2 = h^2$, so $b = \mathbf{h}$. Now, let's check the middle term. The term $14gh$ should be $2ab$. So, $2(7g)(h) = 14gh$. This matches the given expression. So, the expression fits the identity with $a=7g$ and $b=h$.

$49g^2 + 14gh + h^2$

$= (7g)^2 + 2(7g)(h) + (h)^2$

$\boxed{(7g + h)^2}$

Step 5 — Factor $64u^2 + 121v^2 + 4w^2 - 176uv - 32uw + 44vw$

This expression has three squared terms and three product terms. This suggests the identity $(x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx$. Let's identify $x$, $y$, and $z$. We have $x^2 = 64u^2$, so $x = \mathbf{8u}$. We have $y^2 = 121v^2$. We have $z^2 = 4w^2$. Now, let's look at the product terms to find the signs. The term $-176uv$ is $2xy$. So, $2(8u)y = -176uv$. This means $16uy = -176uv$, so $y = \mathbf{-11v}$. The term $-32uw$ is $2xz$. So, $2(8u)z = -32uw$. This means $16uz = -32uw$, so $z = \mathbf{-2w}$. Let's check the last product term $2yz$. $2(-11v)(-2w) = 44vw$. This matches the given expression. So, the expression fits the identity with $x=8u$, $y=-11v$, and $z=-2w$.

$64u^2 + 121v^2 + 4w^2 - 176uv - 32uw + 44vw$

$= (8u)^2 + (-11v)^2 + (-2w)^2 + 2(8u)(-11v) + 2(-11v)(-2w) + 2(8u)(-2w)$

$\boxed{(8u - 11v - 2w)^2}$

Answer

(i) $(3a - b + 2c)^2$ (ii) $(4s - 5t)^2$ (iii) $(r - 7)(r + 6)$ (iv) $(7g + h)^2$ (v) $(8u - 11v - 2w)^2$