How to factor binomials?

Let us consider a binomial expression

For example,

x²+(a+b)x+ab

x²+ax+bx+ab

x(x+a)+b(x+a)

(x+a)(x+b)

Therefore, the above binomial expression has two factors (x+a) and (x+b).

Now by numericals

Consider, x²+12x+35

Here, (x+5) and (x+7) are the factors of the binomial expression x²+12x+35.

Binomials are two-term expressions connected with a plus sign or minus sign, such as {ax+b}ax+b. The first term will always include a variable, while the second term may or may not. 1st: factoring is when you break a large number down into it's simplest divisible parts. Each one of these parts is called a "factor." So, for example, the number 6 can be evenly divided by four different numbers. 2nd:  A binomial is simply the addition or subtraction of two numbers, at least one of which contains a variable. Sometimes these variables have exponents, like x^5 or 5y^6. 3rd :When first factoring binomials, it can help to reorder equations with ascending variable terms. This means you find the highest possible number that both parts of the binomial are divisible by. If you're struggling, simply factor both numbers on their own, then see what the highest matching number is.

Step-by-step explanation:

part a

answer: a reasonable domain is where n is a real number. so n can be between 0 and 10. both endpoints are included.

work shown:

n is the number of days after the experiment starts. the smallest n can be is n = 0 which means that 0 days have gone by, and we're at the start. to find out how large n should be, then replace f(n) with 16.13 and solve for n. use logarithms to isolate the exponent.

f(n) = 12*(1.03)^n

16.13 = 12*(1.03)^n

16.13/12 = (1.03)^n

1.34416666666667 = (1.03)^n

(1.03)^n = 1.34416666666667

log[ (1.03)^n ] = log[ 1.34416666666667 ]

n*log[ 1.03 ] = log[ 1.34416666666667 ]

n = log[ 1.34416666666667 ]/log[ 1.03 ]

n = 10.0062999823929

this rounds to n = 10 which is fairly close but not 100% perfect. so this is the largest n can be.

note: if you plug n = 10 into f(n), you'll get roughly 16.127 which rounds to 16.13 (this comes up again in part c)

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part b

answer: the y intercept is 12. it represents the starting height of the plant in cm.

work shown:

plug n = 0 into the f(n) function. simplify

f(n) = 12*(1.03)^n

f(0) = 12*(1.03)^0

f(0) = 12*(1)

f(0) = 12

on day n = 0, aka the starting point, the height f(n) is 12 cm

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part c

answer: the average rate of change is approximately 0.43061036458991 (round however you need to). this represents the average growth rate from day n = 3 to day n = 10. so the plant grew roughly 0.43 cm per day during this timespan, assuming you round to 2 decimal places.

work shown:

compute f(3)

f(n) = 12*(1.03)^n

f(3) = 12*(1.03)^3

f(3) = 13.112724 < we'll use this later

compute f(10)

f(n) = 12*(1.03)^n

f(10) = 12*(1.03)^10

f(10) = 16.1269965521294 < we'll use this later

now use the formula below with a = 3 and b = 10

aroc = average rate of change

aroc = [ f(b) - f(a) ]/[ b - a ]

aroc = [ f(10) - f(3) ]/[ 10 - 3 ]

aroc = (16.1269965521294 - 13.112724)/(10 - 3)

aroc = 3.0142725521294/7

aroc = 0.43061036458991

round this however you need to

note: the plant grew approximately 3.01 cm over 7 days, so roughly 0.43 cm per day is the average growth rate (if you were to round to 2 decimal places).

v = 1/3 a^2h

step-by-step explanation:

the volume is 1/3 a^2h.