Math / Algebra

Question1. Multiply the following binomials. Simplify where possible. $\text { a. } \begin{aligned} & (3-x)(12-x) \\ & (3)(12)+(3)(-x)+(-x)(12)+(-x)(-x)= \\ = & 36-3 x-12 x+x= \\ = & 36-14 x \end{aligned}$ b. $(9 a b-6 b)(14 b+3 a)$

Studdy Solution

STEP 1

What is this asking? We're multiplying pairs of expressions, each with two terms!
This is like expanding or FOILing. Watch out! Don't forget to distribute both terms in the first parentheses to both terms in the second parentheses.
Also, be careful with those negative signs; they can be sneaky!

STEP 2

1. Multiply the first pair.
2. Multiply the second pair.

STEP 3

Let's distribute the first term of $(3-x)$ , which is $3$ , to each term in $(12-x)$ .
This gives us $3 \cdot 12$ and $3 \cdot (-x)$ .
So, we have $36$ and $-3x$ .

STEP 4

Now, let's distribute the second term of $(3-x)$ , which is $-x$ , to each term in $(12-x)$ .
This gives us $(-x) \cdot 12$ and $(-x) \cdot (-x)$ .
So, we have $-12x$ and $x^2$ .

STEP 5

Putting it all together, we have $36 - 3x - 12x + x^2$ .
Now, we can combine those x terms! $-3x$ plus $-12x$ gives us $-15x$ .
So, our simplified expression is $x^2 - 15x + 36$ .

STEP 6

Alright, time to tackle $(9ab - 6b)(14b + 3a)$ !
Let's distribute $9ab$ to both terms in the second parentheses: $(9ab)(14b)$ gives us $126ab^2$ , and $(9ab)(3a)$ gives us $27a^2b$ .

STEP 7

Now, distribute $-6b$ to both terms in the second parentheses: $(-6b)(14b)$ gives us $-84b^2$ , and $(-6b)(3a)$ gives us $-18ab$ .

STEP 8

Let's put all the pieces together: $126ab^2 + 27a^2b - 84b^2 - 18ab$ .
We can combine the terms $126ab^2$ and $-18ab$ by adding to zero to get $108ab^2$ .
So our final expression is $27a^2b + 108ab^2 - 84b^2$ .

STEP 9

a. $(3-x)(12-x) = x^2 - 15x + 36$ b. $(9ab - 6b)(14b + 3a) = 27a^2b + 108ab^2 - 84b^2$

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Question1. Multiply the following binomials. Simplify where possible. $\text { a. } \begin{aligned} & (3-x)(12-x) \\ & (3)(12)+(3)(-x)+(-x)(12)+(-x)(-x)= \\ = & 36-3 x-12 x+x= \\ = & 36-14 x \end{aligned}$ b. $(9 a b-6 b)(14 b+3 a)$

Studdy Solution

STEP 1

What is this asking? We're multiplying pairs of expressions, each with two terms!
This is like expanding or FOILing. Watch out! Don't forget to distribute both terms in the first parentheses to both terms in the second parentheses.
Also, be careful with those negative signs; they can be sneaky!

STEP 2

1. Multiply the first pair.
2. Multiply the second pair.

STEP 3

Let's distribute the first term of $(3-x)$ , which is $3$ , to each term in $(12-x)$ .
This gives us $3 \cdot 12$ and $3 \cdot (-x)$ .
So, we have $36$ and $-3x$ .

STEP 4

Now, let's distribute the second term of $(3-x)$ , which is $-x$ , to each term in $(12-x)$ .
This gives us $(-x) \cdot 12$ and $(-x) \cdot (-x)$ .
So, we have $-12x$ and $x^2$ .

STEP 5

Putting it all together, we have $36 - 3x - 12x + x^2$ .
Now, we can combine those x terms! $-3x$ plus $-12x$ gives us $-15x$ .
So, our simplified expression is $x^2 - 15x + 36$ .

STEP 6

Alright, time to tackle $(9ab - 6b)(14b + 3a)$ !
Let's distribute $9ab$ to both terms in the second parentheses: $(9ab)(14b)$ gives us $126ab^2$ , and $(9ab)(3a)$ gives us $27a^2b$ .

STEP 7

Now, distribute $-6b$ to both terms in the second parentheses: $(-6b)(14b)$ gives us $-84b^2$ , and $(-6b)(3a)$ gives us $-18ab$ .

STEP 8

Let's put all the pieces together: $126ab^2 + 27a^2b - 84b^2 - 18ab$ .
We can combine the terms $126ab^2$ and $-18ab$ by adding to zero to get $108ab^2$ .
So our final expression is $27a^2b + 108ab^2 - 84b^2$ .

STEP 9

a. $(3-x)(12-x) = x^2 - 15x + 36$ b. $(9ab - 6b)(14b + 3a) = 27a^2b + 108ab^2 - 84b^2$

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Studdy Solution

STEP 1

What is this asking? We're multiplying pairs of expressions, each with two terms!This is like expanding or FOILing. Watch out! Don't forget to distribute *both* terms in the first parentheses to *both* terms in the second parentheses.Also, be careful with those negative signs; they can be sneaky!

STEP 2

1. Multiply the first pair.2. Multiply the second pair.

STEP 3

Let's **distribute** the first term of (3−x)(3-x)(3−x), which is 333, to each term in (12−x)(12-x)(12−x).This gives us 3⋅123 \cdot 123⋅12 and 3⋅(−x)3 \cdot (-x)3⋅(−x).So, we have 363636 and −3x-3x−3x.

STEP 4

Now, let's **distribute** the second term of (3−x)(3-x)(3−x), which is −x-x−x, to each term in (12−x)(12-x)(12−x).This gives us (−x)⋅12(-x) \cdot 12(−x)⋅12 and (−x)⋅(−x)(-x) \cdot (-x)(−x)⋅(−x).So, we have −12x-12x−12x and x2x^2x2.

STEP 5

Putting it all together, we have 36−3x−12x+x236 - 3x - 12x + x^236−3x−12x+x2.Now, we can **combine** those *x* terms! −3x-3x−3x plus −12x-12x−12x gives us −15x-15x−15x.So, our simplified expression is x2−15x+36x^2 - 15x + 36x2−15x+36.

STEP 6

Alright, time to tackle (9ab−6b)(14b+3a)(9ab - 6b)(14b + 3a)(9ab−6b)(14b+3a)!Let's **distribute** 9ab9ab9ab to both terms in the second parentheses: (9ab)(14b)(9ab)(14b)(9ab)(14b) gives us 126ab2126ab^2126ab2, and (9ab)(3a)(9ab)(3a)(9ab)(3a) gives us 27a2b27a^2b27a2b.

STEP 7

Now, **distribute** −6b-6b−6b to both terms in the second parentheses: (−6b)(14b)(-6b)(14b)(−6b)(14b) gives us −84b2-84b^2−84b2, and (−6b)(3a)(-6b)(3a)(−6b)(3a) gives us −18ab-18ab−18ab.

STEP 8

STEP 9

a. (3−x)(12−x)=x2−15x+36(3-x)(12-x) = x^2 - 15x + 36(3−x)(12−x)=x2−15x+36 b. (9ab−6b)(14b+3a)=27a2b+108ab2−84b2(9ab - 6b)(14b + 3a) = 27a^2b + 108ab^2 - 84b^2(9ab−6b)(14b+3a)=27a2b+108ab2−84b2

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Studdy solves anything!

Start learning now

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What is this asking? We're multiplying pairs of expressions, each with two terms!
This is like expanding or FOILing. Watch out! Don't forget to distribute both terms in the first parentheses to both terms in the second parentheses.
Also, be careful with those negative signs; they can be sneaky!

1. Multiply the first pair.
2. Multiply the second pair.

Let's distribute the first term of $(3-x)$ , which is $3$ , to each term in $(12-x)$ .
This gives us $3 \cdot 12$ and $3 \cdot (-x)$ .
So, we have $36$ and $-3x$ .

Now, let's distribute the second term of $(3-x)$ , which is $-x$ , to each term in $(12-x)$ .
This gives us $(-x) \cdot 12$ and $(-x) \cdot (-x)$ .
So, we have $-12x$ and $x^2$ .

Putting it all together, we have $36 - 3x - 12x + x^2$ .
Now, we can combine those x terms! $-3x$ plus $-12x$ gives us $-15x$ .
So, our simplified expression is $x^2 - 15x + 36$ .

Alright, time to tackle $(9ab - 6b)(14b + 3a)$ !
Let's distribute $9ab$ to both terms in the second parentheses: $(9ab)(14b)$ gives us $126ab^2$ , and $(9ab)(3a)$ gives us $27a^2b$ .

Now, distribute $-6b$ to both terms in the second parentheses: $(-6b)(14b)$ gives us $-84b^2$ , and $(-6b)(3a)$ gives us $-18ab$ .

a. $(3-x)(12-x) = x^2 - 15x + 36$ b. $(9ab - 6b)(14b + 3a) = 27a^2b + 108ab^2 - 84b^2$