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3 years ago

Multiply x^2+5•x^3+7 I’m getting x^5+5x^3+7x^2+35 But I’m not sure if I’m right

Mathematics

1 answer:

user100 [1]3 years ago

6 0

Answer:

x⁵ + 5x³ + 7x² + 35

Step-by-step explanation:

(x² + 5) (x³ + 7)

To distribute binomials (expressions with two terms), you can use the FOIL method (First, Outer, Inner, Last):

x⁵ + 7x² + 5x³ + 35

To put into standard form, rearrange from highest power to lowest power:

x⁵ + 5x³ + 7x² + 35

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Prove or disprove (from i=0 to n) sum([2i]^4) <= (4n)^4. If true use induction, else give the smallest value of n that it doe

ddd [48]

Answer:

The statement is true for every n between 0 and 77 and it is false for $n\geq 78$

Step-by-step explanation:

First, observe that, for n=0 and n=1 the statement is true:

For n=0: $\sum^{n}_{i=0} (2i)^4=0 \leq 0=(4n)^4$

For n=1: $\sum^{n}_{i=0} (2i)^4=16 \leq 256=(4n)^4$

From this point we will assume that $n\geq 2$

As we can see, $\sum^{n}_{i=0} (2i)^4=\sum^{n}_{i=0} 16i^4=16\sum^{n}_{i=0} i^4$ and $(4n)^4=256n^4$ . Then,

$\sum^{n}_{i=0} (2i)^4 \leq(4n)^4 \iff \sum^{n}_{i=0} i^4 \leq 16n^4$

Now, we will use the formula for the sum of the first 4th powers:

$\sum^{n}_{i=0} i^4=\frac{n^5}{5} +\frac{n^4}{2} +\frac{n^3}{3}-\frac{n}{30}=\frac{6n^5+15n^4+10n^3-n}{30}$

Therefore:

$\sum^{n}_{i=0} i^4 \leq 16n^4 \iff \frac{6n^5+15n^4+10n^3-n}{30} \leq 16n^4 \\\\ \iff 6n^5+10n^3-n \leq 465n^4 \iff 465n^4-6n^5-10n^3+n\geq 0$

and, because $n \geq 0$ ,

$465n^4-6n^5-10n^3+n\geq 0 \iff n(465n^3-6n^4-10n^2+1)\geq 0 \\\iff 465n^3-6n^4-10n^2+1\geq 0 \iff 465n^3-6n^4-10n^2\geq -1\\\iff n^2(465n-6n^2-10)\geq -1$

Observe that, because $n \geq 2$ and is an integer,

$n^2(465n-6n^2-10)\geq -1 \iff 465n-6n^2-10 \geq 0 \iff n(465-6n) \geq 10\\\iff 465-6n \geq 0 \iff n \leq \frac{465}{6}=\frac{155}{2}=77.5$

In concusion, the statement is true if and only if n is a non negative integer such that $n\leq 77$

So, 78 is the smallest value of n that does not satisfy the inequality.

Note: If you compute $(4n)^4- \sum^{n}_{i=0} (2i)^4$ for 77 and 78 you will obtain:

$(4n)^4- \sum^{n}_{i=0} (2i)^4=53810064$

$(4n)^4- \sum^{n}_{i=0} (2i)^4=-61754992$

7 0

3 years ago

At a school picnic, your teacher asks you to mark a field every 10 yards so students can play football. The teacher accidentally

Oxana [17]

Approximately 9 meters are in 10 yards, (as there is approximately 1 meter in a yard, .9144 to be exact.)

3 0

3 years ago

Does someone know the answer

seropon [69]

11 square. units × 7 square. units × 1/2= B. 38.5 square. units.

7 0

3 years ago

Please help i dont understand :(

saul85 [17]

Answer:306

Step-by-step explanation:

since the width of the bottom square is 10, you split it between the triangles on top. You have to do this because 7 is not the whole width of the two triangles. So then the width of the triangles would be 12, 12 times 9 is 108, you'd have 216 total for both triangles and then the area of the square is 90, 90 plus 216 is 306.

6 0

3 years ago

You receive an 11% commission on all the sales you make. This month, you made a sale of $45,050, what is your commission?

poizon [28]

Answer:

4,955.50

Step-by-step explanation:

45050x.11=4955.50

4 0

3 years ago