Write the following as a product of linear factors:

a. $ x^5 + 5x^4 + 10x^3 + 10x^2 + 5x + 1 $

b. $ x^7 + 7x^6 + 21x^5 + 35x^4 + 35x^3 + 21x^2 + 7x + 1 $

Write the following in standard form:

c. $ (x + 1)^6 $

d. $ (x + 1)^8 $

Now, see if you can write the following as a product of linear factors. It might be helpful to write $ (2x + 3)^3 $ in standard form and look for a pattern:

$ 81x^4 + 216x^3 + 216x^2 + 96x + 16 $

To write a polynomial as a product of linear factors, we need to find the roots of the polynomial. By factoring out the middle term and observing symmetry, we can write the polynomials in the desired form. The expansion of (2x + 3)³ can be converted to standard form and used to find 81x⁴ + 216x³ + 216x² + 96x + 16 as a product of linear factors.

To write a polynomial as a product of linear factors, we need to find the roots of the polynomial.

For polynomial a, 1x⁵ + 5x⁴ + 10x³ + 10x² + 5x + 1, we can observe that the polynomial is symmetric and the roots occur in pairs.

By factoring out (x + 1) in the middle term, we can rewrite the polynomial as (x + 1)(x⁴ + 4x³ + 6x² + 4x + 1).

For polynomial b, 1x⁷ + 7x⁶ + 21x⁵ + 35x⁴ + 35x³ + 21x² + 7x + 1, we can observe that the polynomial is symmetric about the middle term.

We can rewrite the polynomial as (x + 1)(x⁶ + 6x⁵ + 15x⁴ + 20x³ + 15x² + 6x + 1).

To write polynomial (2x + 3)³ in standard form, we can apply the binomial theorem.

The expansion of (2x + 3)³ gives us 8x³ + 36x² + 54x + 27.

By rearranging the terms, we can write the polynomial 81x⁴ + 216x³ + 216x² + 96x + 16 as (3x + 2)⁴.

Learn more about Polynomials here:

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