Question

Multiplying polynomials and simplifying expressions

Answer 1

Given:

`(y-4 x)\left(y^(2)+4 y+16\right)`

Resulting form =
`y^(3)+4 y^(2)+a y-4 x y^(2)-a x y-64 x`

To find:

The value of a in the polynomial.

Solution:

`(y-4 x)\left(y^(2)+4 y+16\right)`

Using distributive property:

`a(b+c)=a b+a c`

`(y-4 x)(y^(2)+4 y+16)=y(y^(2)+4 y+16)-4 x(y^(2)+4 y+16)`

Now multiply each of the first term with each of the second term.

`=y y^(2)+y \cdot 4 y+y \cdot 16+(-4 x) y^(2)+(-4 x) \cdot 4 y+(-4 x) \cdot 16`

Applying plus minus rule:
`+(-a)=-a`

`=y^(2) \cdot y+4 y\cdot y+16 y-4 y^(2) x-4 \cdot 4 y x-4 \cdot 16 x`

Apply the exponent rule:
`a^(b) \cdot a^(c)=a^(b+c)`

`=y^(3)+4 y^2+16 y-4 y^(2) x-16 y x-64 x`

Let us compare this with the given resulting form:

`=y^(3)+4 y^(2)+a y-4 x y^(2)-a x y-64 x`

On comparing above two expression, we get a = 16.

The value of a in the polynomial is 16.