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2 years ago

(6y + 3) minus (3y + 6) when y=7

Mathematics

2 answers:

Yuliya22 [10]2 years ago

8 0

Answer:

72 I think, I don't rly know use a calculator lol

never [62]2 years ago

4 0

Answer:

Step-by-step explanation:

((((2•3y3) - 22y2) - 3y) - —) - 2

STEP

Rewriting the whole as an Equivalent Fraction

4.1 Subtracting a fraction from a whole

Rewrite the whole as a fraction using y as the denominator :

6y3 - 4y2 - 3y (6y3 - 4y2 - 3y) • y

6y3 - 4y2 - 3y = —————————————— = ————————————————————

1 y

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

STEP

Pulling out like terms

5.1 Pull out like factors :

6y3 - 4y2 - 3y = y • (6y2 - 4y - 3)

Trying to factor by splitting the middle term

5.2 Factoring 6y2 - 4y - 3

The first term is, 6y2 its coefficient is 6 .

The middle term is, -4y its coefficient is -4 .

The last term, "the constant", is -3

Step-1 : Multiply the coefficient of the first term by the constant 6 • -3 = -18

Step-2 : Find two factors of -18 whose sum equals the coefficient of the middle term, which is -4 .

-18 + 1 = -17

-9 + 2 = -7

-6 + 3 = -3

-3 + 6 = 3

-2 + 9 = 7

-1 + 18 = 17

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Adding fractions that have a common denominator :

5.3 Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

y • (6y2-4y-3) • y - (6) 6y4 - 4y3 - 3y2 - 6

———————————————————————— = ———————————————————

y y

Equation at the end of step

(6y4 - 4y3 - 3y2 - 6)

————————————————————— - 2

STEP

Rewriting the whole as an Equivalent Fraction :

6.1 Subtracting a whole from a fraction

Rewrite the whole as a fraction using y as the denominator :

2 2 • y

2 = — = —————

1 y

Checking for a perfect cube :

6.2 6y4 - 4y3 - 3y2 - 6 is not a perfect cube

Trying to factor by pulling out :

6.3 Factoring: 6y4 - 4y3 - 3y2 - 6

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: -3y2 - 6

Group 2: 6y4 - 4y3

Pull out from each group separately :

Group 1: (y2 + 2) • (-3)

Group 2: (3y - 2) • (2y3)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

6.4 Find roots (zeroes) of : F(y) = 6y4 - 4y3 - 3y2 - 6

Polynomial Roots Calculator is a set of methods aimed at finding values of y for which F(y)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers y which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 6 and the Trailing Constant is -6.

The factor(s) are:

of the Leading Coefficient : 1,2 ,3 ,6

of the Trailing Constant : 1 ,2 ,3 ,6

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 1.00

-1 2 -0.50 -5.88

-1 3 -0.33 -6.11

-1 6 -0.17 -6.06

-2 1 -2.00 110.00

Note - For tidiness, printing of 13 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

6.5 Adding up the two equivalent fractions

(6y4-4y3-3y2-6) - (2 • y) 6y4 - 4y3 - 3y2 - 2y - 6

————————————————————————— = ————————————————————————

y y

Polynomial Roots Calculator :

6.6 Find roots (zeroes) of : F(y) = 6y4 - 4y3 - 3y2 - 2y - 6

See theory in step 6.4

In this case, the Leading Coefficient is 6 and the Trailing Constant is -6.

The factor(s) are:

of the Leading Coefficient : 1,2 ,3 ,6

of the Trailing Constant : 1 ,2 ,3 ,6

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 3.00

-1 2 -0.50 -4.88

-1 3 -0.33 -5.44

-1 6 -0.17 -5.73

-2 1 -2.00 114.00

Note - For tidiness, printing of 13 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Final result :

6y4 - 4y3 - 3y2 - 2y - 6

————————————————————————

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Musya8 [376]

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nevsk [136]

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$\displaystyle\int e^{-x^2/2}\,\mathrm dx$

and that your question is whether the substitution $u=\dfrac x{\sqrt2}$ would work. Well, let's check it out:

$u=\dfrac x{\sqrt2}\implies\mathrm du=\dfrac{\mathrm dx}{\sqrt2}$
$\implies\displaystyle\int e^{-x^2/2}\,\mathrm dx=\sqrt2\int e^{-(\sqrt2\,u)^2/2}\,\mathrm du$
$=\displaystyle\sqrt2\int e^{-u^2}\,\mathrm du$

which essentially brings us to back to where we started. (The substitution only served to remove the scale factor in the exponent.)

What if we tried $u=\sqrt t$ next? Then $\mathrm du=\dfrac{\mathrm dt}{2\sqrt t}$ , giving

$=\displaystyle\frac1{\sqrt2}\int \frac{e^{-(\sqrt t)^2}}{\sqrt t}\,\mathrm dt=\frac1{\sqrt2}\int\frac{e^{-t}}{\sqrt t}\,\mathrm dt$

Next you may be tempted to try to integrate this by parts, but that will get you nowhere.

So how to deal with this integral? The answer lies in what's called the "error function" defined as

$\mathrm{erf}(x)=\displaystyle\frac2{\sqrt\pi}\int_0^xe^{-t^2}\,\mathrm dt$

By the fundamental theorem of calculus, taking the derivative of both sides yields

$\dfrac{\mathrm d}{\mathrm dx}\mathrm{erf}(x)=\dfrac2{\sqrt\pi}e^{-x^2}$

and so the antiderivative would be

$\displaystyle\int e^{-x^2/2}\,\mathrm dx=\sqrt{\frac\pi2}\mathrm{erf}\left(\frac x{\sqrt2}\right)$

The takeaway here is that a new function (i.e. not some combination of simpler functions like regular exponential, logarithmic, periodic, or polynomial functions) is needed to capture the antiderivative.

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