Just 5 questions! plz answer! will give 10 points! thanks!?

Question

Use three or more sentences to describe how you identify whether or not a polynomial is a difference of two squares. Then, factor the polynomial 4x2-36.

Use three or more sentences to describe how you identify whether or not a polynomial is a perfect square trinomial. Then, factor the polynomial x2-40x+400.

Consider the factoring process that has been illustrated below. Use three or more sentences to describe the process used to factor this polynomial.

3y3+4y2-12y-16

=y2(3y+4)-4(3y+4)

=(3y+4)(y2-4)

=(3y+4)(y+2)(y-2)

Use three or more sentences to describe the process you would follow to completely factor the polynomial 3x6+24.

Honors Only: Look at the equation y10n - 2y5n+1 and determine if it fits the conditions of one of the special cases listed above. Explain why the trinomial is or is not a special case and describe your process for factoring it completely. Be sure to use three or more sentences in your explanation. If the polynomial can be factored, then factor it showing your work.

Answer 1

Hope these help:

Note: The right side of each equation below is the factored form.

Difference of Perfect Squares

x^2 - y^2 = (x-y) (x+y)

Perfect Square trinomal

(x + y)^2 = x^2 + 2xy + y^2

Remember:

A number is a perfect square, if the square root of the number is an integer.

A variable is a perfect square, if the square root of the variable is a variable to an integer power.

Answer 2

Case I:

First Check whether the each term of given polynomial is sqaured or not . If u find that it is a square then apply the rule (a^2 -b^2) = (a + b)(a - b)

Look for ur example:

4x^2 - 36

Here 4x^2 can be written as (2x)^2 &

36 can be written as (6)^2

so now apply a^2 - b^2 rule.

so we have,

4x^2 - 36

=(2x)^2 - (6)^2

=(2x + 6)(2x - 6) [Here a = 2x & b = 6]

Case II:

First check whether the given polynomial can be expresses in the manner: (a^2 - 2ab + b^2) or (a^2 + 2ab + b^2).

Look ur example:

x^2 - 4x + 400

first check x^2 = (x)^2 & 400 = (20)^2

so, 2.x.20 = 40x

so we can that the given polynomial can be expressed as a perfect square.

Look, x^2 - 40x + 400

= (x)^2 - 2.x.20 + (20)^2

=(x - 20)^2 => which is a perfect square.

Case III: This type of factorisation is called Vanishing Fcator.

3y^3 - 4y^2 - 12y - 16

first  we assumed that for what value of y, the value of the given polynomial is zero.

if we take y = 2 then,

3.(2)^3 + 4.(2)^2 - 12.2 - 16

= 3.8 + 4.4 - 24 - 16

= 24 + 16 - 24 - 16

= 0

Now, we can say that (y - 2) is one of the factor of the polynomial.

Now, 3y^3 + 4y^2 - 12y - 16

= 3y^3 - 6y^2 + 10y^2 - 20y + 8y - 16

= 3y^2(y - 2) + 10y(y - 2) +8(y - 2)

= (y - 2)(3y^2 + 10y + 8)

=(y - 2)(3y^2 + 6y + 4y + 8)

= (y - 2){3y(y + 2) + 4 (y + 2)}

= (y - 2)(y + 2)(3y + 4) <==ANSWER

If we take y = - 2 or y = - 4/3 , then also the value of given polynomial will be zero. then we can say that (y + 2) or (3y + 4) is one of the factor of the given polynomial.

case IV:

3x^6 + 24

=3 (x^6 + 8)

=3{(x^2)^3 + (2)^3}

=3(x^2 + 3){(x^2)^2 - x^2.2 + (2)^2}

= 3(x^2 + 3)(x^4 - 2x^2 + 4) <==ANSWER

Rule: a^3 + b^3 = (a + b)(a^2 - ab + b^2)

Case V:

y^10n - 2y^5n + 1

= y^10n - y^5n - y^5n + 1

= y^5n(y^5n -1) - 1(y^5n - 1)

= (y^5n - 1)(y^5n - 1)

again this can be express as a perfect square.

y^10n - 2y^5n + 1

=(y^5n)^2 - 2.y^5n.1 + (1)^2

= (y^5n - 1)^2

Hope this helps u :)